Dirichlet series
| L(s) = 1 | + 5·2-s + 6·4-s + 4·5-s − 14·7-s − 11·8-s + 20·10-s + 3·11-s − 13·13-s − 70·14-s − 35·16-s + 5·17-s + 21·19-s + 24·20-s + 15·22-s + 10·23-s − 25·25-s − 65·26-s − 84·28-s + 15·29-s + 33·31-s − 20·32-s + 25·34-s − 56·35-s − 29·37-s + 105·38-s − 44·40-s + 41-s + ⋯ |
| L(s) = 1 | + 3.53·2-s + 3·4-s + 1.78·5-s − 5.29·7-s − 3.88·8-s + 6.32·10-s + 0.904·11-s − 3.60·13-s − 18.7·14-s − 8.75·16-s + 1.21·17-s + 4.81·19-s + 5.36·20-s + 3.19·22-s + 2.08·23-s − 5·25-s − 12.7·26-s − 15.8·28-s + 2.78·29-s + 5.92·31-s − 3.53·32-s + 4.28·34-s − 9.46·35-s − 4.76·37-s + 17.0·38-s − 6.95·40-s + 0.156·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 7^{14} \cdot 127^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{28} \cdot 7^{14} \cdot 127^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(3^{28} \cdot 7^{14} \cdot 127^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.88755\times 10^{25}\) |
| Root analytic conductor: | \(7.99301\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 3^{28} \cdot 7^{14} \cdot 127^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(213.4500480\) |
| \(L(\frac12)\) | \(\approx\) | \(213.4500480\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 7 | \( ( 1 + T )^{14} \) | |
| 127 | \( ( 1 + T )^{14} \) | |
| good | 2 | \( 1 - 5 T + 19 T^{2} - 27 p T^{3} + 17 p^{3} T^{4} - 151 p T^{5} + 627 T^{6} - 1213 T^{7} + 2245 T^{8} - 3945 T^{9} + 3333 p T^{10} - 10741 T^{11} + 8349 p T^{12} - 6219 p^{2} T^{13} + 8973 p^{2} T^{14} - 6219 p^{3} T^{15} + 8349 p^{3} T^{16} - 10741 p^{3} T^{17} + 3333 p^{5} T^{18} - 3945 p^{5} T^{19} + 2245 p^{6} T^{20} - 1213 p^{7} T^{21} + 627 p^{8} T^{22} - 151 p^{10} T^{23} + 17 p^{13} T^{24} - 27 p^{12} T^{25} + 19 p^{12} T^{26} - 5 p^{13} T^{27} + p^{14} T^{28} \) |
| 5 | \( 1 - 4 T + 41 T^{2} - 6 p^{2} T^{3} + 174 p T^{4} - 2908 T^{5} + 12492 T^{6} - 38021 T^{7} + 133996 T^{8} - 370873 T^{9} + 1127299 T^{10} - 567203 p T^{11} + 7634651 T^{12} - 17439737 T^{13} + 42178904 T^{14} - 17439737 p T^{15} + 7634651 p^{2} T^{16} - 567203 p^{4} T^{17} + 1127299 p^{4} T^{18} - 370873 p^{5} T^{19} + 133996 p^{6} T^{20} - 38021 p^{7} T^{21} + 12492 p^{8} T^{22} - 2908 p^{9} T^{23} + 174 p^{11} T^{24} - 6 p^{13} T^{25} + 41 p^{12} T^{26} - 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 11 | \( 1 - 3 T + 82 T^{2} - 256 T^{3} + 3350 T^{4} - 10553 T^{5} + 92717 T^{6} - 284301 T^{7} + 1964384 T^{8} - 5713164 T^{9} + 33629114 T^{10} - 91772019 T^{11} + 477433889 T^{12} - 1214718248 T^{13} + 5707030766 T^{14} - 1214718248 p T^{15} + 477433889 p^{2} T^{16} - 91772019 p^{3} T^{17} + 33629114 p^{4} T^{18} - 5713164 p^{5} T^{19} + 1964384 p^{6} T^{20} - 284301 p^{7} T^{21} + 92717 p^{8} T^{22} - 10553 p^{9} T^{23} + 3350 p^{10} T^{24} - 256 p^{11} T^{25} + 82 p^{12} T^{26} - 3 p^{13} T^{27} + p^{14} T^{28} \) | |
| 13 | \( 1 + p T + 14 p T^{2} + 1585 T^{3} + 13608 T^{4} + 92167 T^{5} + 46685 p T^{6} + 3422512 T^{7} + 18744862 T^{8} + 91352922 T^{9} + 433195838 T^{10} + 1864962208 T^{11} + 7828834193 T^{12} + 30141395677 T^{13} + 113300440326 T^{14} + 30141395677 p T^{15} + 7828834193 p^{2} T^{16} + 1864962208 p^{3} T^{17} + 433195838 p^{4} T^{18} + 91352922 p^{5} T^{19} + 18744862 p^{6} T^{20} + 3422512 p^{7} T^{21} + 46685 p^{9} T^{22} + 92167 p^{9} T^{23} + 13608 p^{10} T^{24} + 1585 p^{11} T^{25} + 14 p^{13} T^{26} + p^{14} T^{27} + p^{14} T^{28} \) | |
| 17 | \( 1 - 5 T + 130 T^{2} - 468 T^{3} + 7643 T^{4} - 20469 T^{5} + 287362 T^{6} - 594229 T^{7} + 479388 p T^{8} - 13821766 T^{9} + 190977908 T^{10} - 284790841 T^{11} + 3876597272 T^{12} - 5340393118 T^{13} + 69746292016 T^{14} - 5340393118 p T^{15} + 3876597272 p^{2} T^{16} - 284790841 p^{3} T^{17} + 190977908 p^{4} T^{18} - 13821766 p^{5} T^{19} + 479388 p^{7} T^{20} - 594229 p^{7} T^{21} + 287362 p^{8} T^{22} - 20469 p^{9} T^{23} + 7643 p^{10} T^{24} - 468 p^{11} T^{25} + 130 p^{12} T^{26} - 5 p^{13} T^{27} + p^{14} T^{28} \) | |
| 19 | \( 1 - 21 T + 374 T^{2} - 4569 T^{3} + 49627 T^{4} - 447130 T^{5} + 3686027 T^{6} - 26871562 T^{7} + 182306081 T^{8} - 1126820515 T^{9} + 6549395980 T^{10} - 35231461213 T^{11} + 179334900367 T^{12} - 851666403346 T^{13} + 202030289050 p T^{14} - 851666403346 p T^{15} + 179334900367 p^{2} T^{16} - 35231461213 p^{3} T^{17} + 6549395980 p^{4} T^{18} - 1126820515 p^{5} T^{19} + 182306081 p^{6} T^{20} - 26871562 p^{7} T^{21} + 3686027 p^{8} T^{22} - 447130 p^{9} T^{23} + 49627 p^{10} T^{24} - 4569 p^{11} T^{25} + 374 p^{12} T^{26} - 21 p^{13} T^{27} + p^{14} T^{28} \) | |
| 23 | \( 1 - 10 T + 197 T^{2} - 1651 T^{3} + 18643 T^{4} - 131352 T^{5} + 1107307 T^{6} - 290175 p T^{7} + 46362563 T^{8} - 244354670 T^{9} + 1483666023 T^{10} - 7040607329 T^{11} + 39336210461 T^{12} - 174731022341 T^{13} + 936647851466 T^{14} - 174731022341 p T^{15} + 39336210461 p^{2} T^{16} - 7040607329 p^{3} T^{17} + 1483666023 p^{4} T^{18} - 244354670 p^{5} T^{19} + 46362563 p^{6} T^{20} - 290175 p^{8} T^{21} + 1107307 p^{8} T^{22} - 131352 p^{9} T^{23} + 18643 p^{10} T^{24} - 1651 p^{11} T^{25} + 197 p^{12} T^{26} - 10 p^{13} T^{27} + p^{14} T^{28} \) | |
| 29 | \( 1 - 15 T + 325 T^{2} - 3542 T^{3} + 45090 T^{4} - 389302 T^{5} + 3709822 T^{6} - 26645384 T^{7} + 208750551 T^{8} - 1291861725 T^{9} + 8811883923 T^{10} - 48651787808 T^{11} + 303330222126 T^{12} - 1554455012568 T^{13} + 9203796607924 T^{14} - 1554455012568 p T^{15} + 303330222126 p^{2} T^{16} - 48651787808 p^{3} T^{17} + 8811883923 p^{4} T^{18} - 1291861725 p^{5} T^{19} + 208750551 p^{6} T^{20} - 26645384 p^{7} T^{21} + 3709822 p^{8} T^{22} - 389302 p^{9} T^{23} + 45090 p^{10} T^{24} - 3542 p^{11} T^{25} + 325 p^{12} T^{26} - 15 p^{13} T^{27} + p^{14} T^{28} \) | |
| 31 | \( 1 - 33 T + 772 T^{2} - 13150 T^{3} + 188546 T^{4} - 2298977 T^{5} + 24968647 T^{6} - 242769108 T^{7} + 2159103944 T^{8} - 567899079 p T^{9} + 133190121022 T^{10} - 935380216908 T^{11} + 6142523519475 T^{12} - 37677429544483 T^{13} + 216850565998226 T^{14} - 37677429544483 p T^{15} + 6142523519475 p^{2} T^{16} - 935380216908 p^{3} T^{17} + 133190121022 p^{4} T^{18} - 567899079 p^{6} T^{19} + 2159103944 p^{6} T^{20} - 242769108 p^{7} T^{21} + 24968647 p^{8} T^{22} - 2298977 p^{9} T^{23} + 188546 p^{10} T^{24} - 13150 p^{11} T^{25} + 772 p^{12} T^{26} - 33 p^{13} T^{27} + p^{14} T^{28} \) | |
| 37 | \( 1 + 29 T + 693 T^{2} + 11721 T^{3} + 4698 p T^{4} + 2176097 T^{5} + 24709960 T^{6} + 250869276 T^{7} + 2356076231 T^{8} + 20315367197 T^{9} + 163971869231 T^{10} + 1232026880055 T^{11} + 8724657176274 T^{12} + 57914924180169 T^{13} + 363474458224772 T^{14} + 57914924180169 p T^{15} + 8724657176274 p^{2} T^{16} + 1232026880055 p^{3} T^{17} + 163971869231 p^{4} T^{18} + 20315367197 p^{5} T^{19} + 2356076231 p^{6} T^{20} + 250869276 p^{7} T^{21} + 24709960 p^{8} T^{22} + 2176097 p^{9} T^{23} + 4698 p^{11} T^{24} + 11721 p^{11} T^{25} + 693 p^{12} T^{26} + 29 p^{13} T^{27} + p^{14} T^{28} \) | |
| 41 | \( 1 - T + 206 T^{2} - 42 T^{3} + 22600 T^{4} + 5720 T^{5} + 1802471 T^{6} + 973585 T^{7} + 116330717 T^{8} + 88412265 T^{9} + 6386509096 T^{10} + 5890073146 T^{11} + 308288137130 T^{12} + 303314192679 T^{13} + 13329630557542 T^{14} + 303314192679 p T^{15} + 308288137130 p^{2} T^{16} + 5890073146 p^{3} T^{17} + 6386509096 p^{4} T^{18} + 88412265 p^{5} T^{19} + 116330717 p^{6} T^{20} + 973585 p^{7} T^{21} + 1802471 p^{8} T^{22} + 5720 p^{9} T^{23} + 22600 p^{10} T^{24} - 42 p^{11} T^{25} + 206 p^{12} T^{26} - p^{13} T^{27} + p^{14} T^{28} \) | |
| 43 | \( 1 + 25 T + 530 T^{2} + 7932 T^{3} + 110412 T^{4} + 1292969 T^{5} + 14382905 T^{6} + 142501427 T^{7} + 1353730590 T^{8} + 11754987450 T^{9} + 98445954790 T^{10} + 763920391599 T^{11} + 5734234384189 T^{12} + 40147595965818 T^{13} + 272318392470142 T^{14} + 40147595965818 p T^{15} + 5734234384189 p^{2} T^{16} + 763920391599 p^{3} T^{17} + 98445954790 p^{4} T^{18} + 11754987450 p^{5} T^{19} + 1353730590 p^{6} T^{20} + 142501427 p^{7} T^{21} + 14382905 p^{8} T^{22} + 1292969 p^{9} T^{23} + 110412 p^{10} T^{24} + 7932 p^{11} T^{25} + 530 p^{12} T^{26} + 25 p^{13} T^{27} + p^{14} T^{28} \) | |
| 47 | \( 1 - 9 T + 350 T^{2} - 3582 T^{3} + 68307 T^{4} - 684242 T^{5} + 9374099 T^{6} - 86334288 T^{7} + 965719113 T^{8} - 8092193530 T^{9} + 77594134130 T^{10} - 590988285775 T^{11} + 4991471177695 T^{12} - 34417211240088 T^{13} + 260228788002610 T^{14} - 34417211240088 p T^{15} + 4991471177695 p^{2} T^{16} - 590988285775 p^{3} T^{17} + 77594134130 p^{4} T^{18} - 8092193530 p^{5} T^{19} + 965719113 p^{6} T^{20} - 86334288 p^{7} T^{21} + 9374099 p^{8} T^{22} - 684242 p^{9} T^{23} + 68307 p^{10} T^{24} - 3582 p^{11} T^{25} + 350 p^{12} T^{26} - 9 p^{13} T^{27} + p^{14} T^{28} \) | |
| 53 | \( 1 - 35 T + 934 T^{2} - 17975 T^{3} + 299033 T^{4} - 4227887 T^{5} + 54144485 T^{6} - 622708390 T^{7} + 6648233679 T^{8} - 65496431709 T^{9} + 608126212712 T^{10} - 5291692221571 T^{11} + 43805743758863 T^{12} - 342748023160961 T^{13} + 2563077085008698 T^{14} - 342748023160961 p T^{15} + 43805743758863 p^{2} T^{16} - 5291692221571 p^{3} T^{17} + 608126212712 p^{4} T^{18} - 65496431709 p^{5} T^{19} + 6648233679 p^{6} T^{20} - 622708390 p^{7} T^{21} + 54144485 p^{8} T^{22} - 4227887 p^{9} T^{23} + 299033 p^{10} T^{24} - 17975 p^{11} T^{25} + 934 p^{12} T^{26} - 35 p^{13} T^{27} + p^{14} T^{28} \) | |
| 59 | \( 1 + 10 T + 389 T^{2} + 3033 T^{3} + 74371 T^{4} + 531008 T^{5} + 10011285 T^{6} + 70337571 T^{7} + 1054894651 T^{8} + 7383068244 T^{9} + 90904277275 T^{10} + 631193665181 T^{11} + 6652047058913 T^{12} + 44814617366759 T^{13} + 421301398427094 T^{14} + 44814617366759 p T^{15} + 6652047058913 p^{2} T^{16} + 631193665181 p^{3} T^{17} + 90904277275 p^{4} T^{18} + 7383068244 p^{5} T^{19} + 1054894651 p^{6} T^{20} + 70337571 p^{7} T^{21} + 10011285 p^{8} T^{22} + 531008 p^{9} T^{23} + 74371 p^{10} T^{24} + 3033 p^{11} T^{25} + 389 p^{12} T^{26} + 10 p^{13} T^{27} + p^{14} T^{28} \) | |
| 61 | \( 1 - T + 410 T^{2} - 890 T^{3} + 87588 T^{4} - 279433 T^{5} + 12931865 T^{6} - 50711894 T^{7} + 1484437808 T^{8} - 6339757701 T^{9} + 139727676318 T^{10} - 597738020304 T^{11} + 11017030143027 T^{12} - 44768426473041 T^{13} + 731807583059774 T^{14} - 44768426473041 p T^{15} + 11017030143027 p^{2} T^{16} - 597738020304 p^{3} T^{17} + 139727676318 p^{4} T^{18} - 6339757701 p^{5} T^{19} + 1484437808 p^{6} T^{20} - 50711894 p^{7} T^{21} + 12931865 p^{8} T^{22} - 279433 p^{9} T^{23} + 87588 p^{10} T^{24} - 890 p^{11} T^{25} + 410 p^{12} T^{26} - p^{13} T^{27} + p^{14} T^{28} \) | |
| 67 | \( 1 + 38 T + 1174 T^{2} + 23934 T^{3} + 422665 T^{4} + 5839509 T^{5} + 71970295 T^{6} + 723216019 T^{7} + 6601917413 T^{8} + 48533153402 T^{9} + 331369870442 T^{10} + 1636952749842 T^{11} + 8113580369697 T^{12} + 14028985989802 T^{13} + 123870851464306 T^{14} + 14028985989802 p T^{15} + 8113580369697 p^{2} T^{16} + 1636952749842 p^{3} T^{17} + 331369870442 p^{4} T^{18} + 48533153402 p^{5} T^{19} + 6601917413 p^{6} T^{20} + 723216019 p^{7} T^{21} + 71970295 p^{8} T^{22} + 5839509 p^{9} T^{23} + 422665 p^{10} T^{24} + 23934 p^{11} T^{25} + 1174 p^{12} T^{26} + 38 p^{13} T^{27} + p^{14} T^{28} \) | |
| 71 | \( 1 - 10 T + 621 T^{2} - 5411 T^{3} + 182283 T^{4} - 1425360 T^{5} + 34456003 T^{6} - 247816664 T^{7} + 4799991247 T^{8} - 32264035049 T^{9} + 530496962459 T^{10} - 3348973491864 T^{11} + 48460938627045 T^{12} - 285739722695784 T^{13} + 3734602641050538 T^{14} - 285739722695784 p T^{15} + 48460938627045 p^{2} T^{16} - 3348973491864 p^{3} T^{17} + 530496962459 p^{4} T^{18} - 32264035049 p^{5} T^{19} + 4799991247 p^{6} T^{20} - 247816664 p^{7} T^{21} + 34456003 p^{8} T^{22} - 1425360 p^{9} T^{23} + 182283 p^{10} T^{24} - 5411 p^{11} T^{25} + 621 p^{12} T^{26} - 10 p^{13} T^{27} + p^{14} T^{28} \) | |
| 73 | \( 1 - 8 T + 547 T^{2} - 3939 T^{3} + 156457 T^{4} - 1057340 T^{5} + 30751475 T^{6} - 195813211 T^{7} + 4575405639 T^{8} - 27362929950 T^{9} + 541643272461 T^{10} - 3022765696423 T^{11} + 52384471961759 T^{12} - 270009836377377 T^{13} + 4193767875322650 T^{14} - 270009836377377 p T^{15} + 52384471961759 p^{2} T^{16} - 3022765696423 p^{3} T^{17} + 541643272461 p^{4} T^{18} - 27362929950 p^{5} T^{19} + 4575405639 p^{6} T^{20} - 195813211 p^{7} T^{21} + 30751475 p^{8} T^{22} - 1057340 p^{9} T^{23} + 156457 p^{10} T^{24} - 3939 p^{11} T^{25} + 547 p^{12} T^{26} - 8 p^{13} T^{27} + p^{14} T^{28} \) | |
| 79 | \( 1 - 26 T + 995 T^{2} - 18888 T^{3} + 423915 T^{4} - 6445491 T^{5} + 108617293 T^{6} - 1396649186 T^{7} + 19434091386 T^{8} - 2770182835 p T^{9} + 2638999284435 T^{10} - 26567340357667 T^{11} + 284690572139722 T^{12} - 2585314414532307 T^{13} + 24906831120291546 T^{14} - 2585314414532307 p T^{15} + 284690572139722 p^{2} T^{16} - 26567340357667 p^{3} T^{17} + 2638999284435 p^{4} T^{18} - 2770182835 p^{6} T^{19} + 19434091386 p^{6} T^{20} - 1396649186 p^{7} T^{21} + 108617293 p^{8} T^{22} - 6445491 p^{9} T^{23} + 423915 p^{10} T^{24} - 18888 p^{11} T^{25} + 995 p^{12} T^{26} - 26 p^{13} T^{27} + p^{14} T^{28} \) | |
| 83 | \( 1 - 30 T + 969 T^{2} - 18676 T^{3} + 365335 T^{4} - 5350552 T^{5} + 79155481 T^{6} - 943348279 T^{7} + 11483465659 T^{8} - 115946714119 T^{9} + 1228033468687 T^{10} - 10957644918809 T^{11} + 107585686563309 T^{12} - 906460220784007 T^{13} + 8857114926688414 T^{14} - 906460220784007 p T^{15} + 107585686563309 p^{2} T^{16} - 10957644918809 p^{3} T^{17} + 1228033468687 p^{4} T^{18} - 115946714119 p^{5} T^{19} + 11483465659 p^{6} T^{20} - 943348279 p^{7} T^{21} + 79155481 p^{8} T^{22} - 5350552 p^{9} T^{23} + 365335 p^{10} T^{24} - 18676 p^{11} T^{25} + 969 p^{12} T^{26} - 30 p^{13} T^{27} + p^{14} T^{28} \) | |
| 89 | \( 1 + 4 T + 780 T^{2} + 1881 T^{3} + 289356 T^{4} + 274990 T^{5} + 69261693 T^{6} - 17925425 T^{7} + 12271686510 T^{8} - 13239275906 T^{9} + 1735603354082 T^{10} - 2547188943427 T^{11} + 202853568948719 T^{12} - 309799827574717 T^{13} + 19768290680155150 T^{14} - 309799827574717 p T^{15} + 202853568948719 p^{2} T^{16} - 2547188943427 p^{3} T^{17} + 1735603354082 p^{4} T^{18} - 13239275906 p^{5} T^{19} + 12271686510 p^{6} T^{20} - 17925425 p^{7} T^{21} + 69261693 p^{8} T^{22} + 274990 p^{9} T^{23} + 289356 p^{10} T^{24} + 1881 p^{11} T^{25} + 780 p^{12} T^{26} + 4 p^{13} T^{27} + p^{14} T^{28} \) | |
| 97 | \( 1 - 15 T + 688 T^{2} - 9074 T^{3} + 232439 T^{4} - 2674739 T^{5} + 50852864 T^{6} - 510202963 T^{7} + 8088855856 T^{8} - 71493024766 T^{9} + 1014684615820 T^{10} - 8074806425495 T^{11} + 108805877810262 T^{12} - 810761204923202 T^{13} + 10784014670664272 T^{14} - 810761204923202 p T^{15} + 108805877810262 p^{2} T^{16} - 8074806425495 p^{3} T^{17} + 1014684615820 p^{4} T^{18} - 71493024766 p^{5} T^{19} + 8088855856 p^{6} T^{20} - 510202963 p^{7} T^{21} + 50852864 p^{8} T^{22} - 2674739 p^{9} T^{23} + 232439 p^{10} T^{24} - 9074 p^{11} T^{25} + 688 p^{12} T^{26} - 15 p^{13} T^{27} + p^{14} T^{28} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.15136016648177539858647178765, −2.13165768668374388688190782431, −1.92596936113355722641966400071, −1.89643180369223521101332761706, −1.75343317186506604922827737621, −1.73245861671160142641334241298, −1.73028949378240316132940158945, −1.47404208304694641534102340694, −1.45232741930047618588852933020, −1.42705695921294308753669049316, −1.42294655210945007922207674817, −1.26828698600340932266300099102, −1.07487207832209839848710178834, −0.993093473973767803921559251980, −0.821406678770432666515286172492, −0.810679312653126279019806097586, −0.77807337219912186035654991719, −0.74200653007821373464211206266, −0.68452272690584216389973978275, −0.47034786938284779619544578725, −0.42916325068170793385182357402, −0.41945031381381770228230049071, −0.33659754326271457986403537229, −0.31230906647531436880550915440, −0.20675756587730571961680992966, 0.20675756587730571961680992966, 0.31230906647531436880550915440, 0.33659754326271457986403537229, 0.41945031381381770228230049071, 0.42916325068170793385182357402, 0.47034786938284779619544578725, 0.68452272690584216389973978275, 0.74200653007821373464211206266, 0.77807337219912186035654991719, 0.810679312653126279019806097586, 0.821406678770432666515286172492, 0.993093473973767803921559251980, 1.07487207832209839848710178834, 1.26828698600340932266300099102, 1.42294655210945007922207674817, 1.42705695921294308753669049316, 1.45232741930047618588852933020, 1.47404208304694641534102340694, 1.73028949378240316132940158945, 1.73245861671160142641334241298, 1.75343317186506604922827737621, 1.89643180369223521101332761706, 1.92596936113355722641966400071, 2.13165768668374388688190782431, 2.15136016648177539858647178765
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.