Dirichlet series
| L(s) = 1 | − 3·3-s − 4·5-s + 8·7-s + 7·9-s − 7·11-s − 11·13-s + 12·15-s + 9·17-s − 2·19-s − 24·21-s + 20·23-s + 6·25-s − 17·27-s + 29-s − 2·31-s + 21·33-s − 32·35-s − 16·37-s + 33·39-s − 11·41-s − 16·43-s − 28·45-s − 10·47-s + 48·49-s − 27·51-s − 9·53-s + 28·55-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.78·5-s + 3.02·7-s + 7/3·9-s − 2.11·11-s − 3.05·13-s + 3.09·15-s + 2.18·17-s − 0.458·19-s − 5.23·21-s + 4.17·23-s + 6/5·25-s − 3.27·27-s + 0.185·29-s − 0.359·31-s + 3.65·33-s − 5.40·35-s − 2.63·37-s + 5.28·39-s − 1.71·41-s − 2.43·43-s − 4.17·45-s − 1.45·47-s + 48/7·49-s − 3.78·51-s − 1.23·53-s + 3.77·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{16} \cdot 11^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(32\) |
| Conductor: | \(2^{48} \cdot 5^{16} \cdot 11^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.39108\times 10^{8}\) |
| Root analytic conductor: | \(1.87441\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((32,\ 2^{48} \cdot 5^{16} \cdot 11^{16} ,\ ( \ : [1/2]^{16} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.049963766\) |
| \(L(\frac12)\) | \(\approx\) | \(1.049963766\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{4} \) | |
| 11 | \( 1 + 7 T + 35 T^{2} + 74 T^{3} - 24 T^{4} - 603 T^{5} + 738 T^{6} + 16446 T^{7} + 88553 T^{8} + 16446 p T^{9} + 738 p^{2} T^{10} - 603 p^{3} T^{11} - 24 p^{4} T^{12} + 74 p^{5} T^{13} + 35 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| good | 3 | \( 1 + p T + 2 T^{2} + 2 T^{3} + p^{2} T^{4} - 2 T^{5} - 38 T^{6} - 32 p T^{7} - 224 T^{8} - 476 T^{9} - 872 T^{10} - 923 T^{11} - 28 p T^{12} + 1702 T^{13} + 7814 T^{14} + 16238 T^{15} + 24007 T^{16} + 16238 p T^{17} + 7814 p^{2} T^{18} + 1702 p^{3} T^{19} - 28 p^{5} T^{20} - 923 p^{5} T^{21} - 872 p^{6} T^{22} - 476 p^{7} T^{23} - 224 p^{8} T^{24} - 32 p^{10} T^{25} - 38 p^{10} T^{26} - 2 p^{11} T^{27} + p^{14} T^{28} + 2 p^{13} T^{29} + 2 p^{14} T^{30} + p^{16} T^{31} + p^{16} T^{32} \) |
| 7 | \( 1 - 8 T + 16 T^{2} + 16 T^{3} - 44 T^{4} + 116 T^{5} - 158 p T^{6} + 234 T^{7} + 10757 T^{8} - 15999 T^{9} + 1380 p T^{10} - 84062 T^{11} + 117927 T^{12} + 148257 T^{13} - 19743 p T^{14} - 25418 p T^{15} - 47876 T^{16} - 25418 p^{2} T^{17} - 19743 p^{3} T^{18} + 148257 p^{3} T^{19} + 117927 p^{4} T^{20} - 84062 p^{5} T^{21} + 1380 p^{7} T^{22} - 15999 p^{7} T^{23} + 10757 p^{8} T^{24} + 234 p^{9} T^{25} - 158 p^{11} T^{26} + 116 p^{11} T^{27} - 44 p^{12} T^{28} + 16 p^{13} T^{29} + 16 p^{14} T^{30} - 8 p^{15} T^{31} + p^{16} T^{32} \) | |
| 13 | \( 1 + 11 T + 41 T^{2} + 64 T^{3} + 44 T^{4} - 503 T^{5} + 1255 T^{6} + 33356 T^{7} + 74317 T^{8} - 108817 T^{9} - 749054 T^{10} - 5432367 T^{11} - 9105150 T^{12} + 46607595 T^{13} - 5926855 p T^{14} - 1188823295 T^{15} - 3786673894 T^{16} - 1188823295 p T^{17} - 5926855 p^{3} T^{18} + 46607595 p^{3} T^{19} - 9105150 p^{4} T^{20} - 5432367 p^{5} T^{21} - 749054 p^{6} T^{22} - 108817 p^{7} T^{23} + 74317 p^{8} T^{24} + 33356 p^{9} T^{25} + 1255 p^{10} T^{26} - 503 p^{11} T^{27} + 44 p^{12} T^{28} + 64 p^{13} T^{29} + 41 p^{14} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 17 | \( 1 - 9 T - 16 T^{2} + 389 T^{3} - 655 T^{4} - 5925 T^{5} + 19742 T^{6} + 39121 T^{7} - 164482 T^{8} - 22403 p T^{9} + 409344 T^{10} + 3609663 T^{11} + 23398512 T^{12} + 60329322 T^{13} - 1416458590 T^{14} - 891314344 T^{15} + 34019499373 T^{16} - 891314344 p T^{17} - 1416458590 p^{2} T^{18} + 60329322 p^{3} T^{19} + 23398512 p^{4} T^{20} + 3609663 p^{5} T^{21} + 409344 p^{6} T^{22} - 22403 p^{8} T^{23} - 164482 p^{8} T^{24} + 39121 p^{9} T^{25} + 19742 p^{10} T^{26} - 5925 p^{11} T^{27} - 655 p^{12} T^{28} + 389 p^{13} T^{29} - 16 p^{14} T^{30} - 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 19 | \( 1 + 2 T - 70 T^{2} - 135 T^{3} + 1701 T^{4} + 2722 T^{5} - 740 T^{6} + 21305 T^{7} - 906739 T^{8} - 1094233 T^{9} + 21790339 T^{10} - 26382512 T^{11} - 160009494 T^{12} + 1525825782 T^{13} - 3404531246 T^{14} - 16913782877 T^{15} + 117170393016 T^{16} - 16913782877 p T^{17} - 3404531246 p^{2} T^{18} + 1525825782 p^{3} T^{19} - 160009494 p^{4} T^{20} - 26382512 p^{5} T^{21} + 21790339 p^{6} T^{22} - 1094233 p^{7} T^{23} - 906739 p^{8} T^{24} + 21305 p^{9} T^{25} - 740 p^{10} T^{26} + 2722 p^{11} T^{27} + 1701 p^{12} T^{28} - 135 p^{13} T^{29} - 70 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 23 | \( ( 1 - 10 T + 192 T^{2} - 1521 T^{3} + 15861 T^{4} - 101094 T^{5} + 739787 T^{6} - 3804023 T^{7} + 21361714 T^{8} - 3804023 p T^{9} + 739787 p^{2} T^{10} - 101094 p^{3} T^{11} + 15861 p^{4} T^{12} - 1521 p^{5} T^{13} + 192 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 29 | \( 1 - T - 7 T^{2} - 100 T^{3} + 654 T^{4} + 10289 T^{5} - 2925 T^{6} - 3617 p T^{7} - 115321 T^{8} + 8390856 T^{9} + 39391395 T^{10} - 170919005 T^{11} - 283495188 T^{12} + 6984126676 T^{13} + 61979011448 T^{14} + 26080709508 T^{15} - 1384555826449 T^{16} + 26080709508 p T^{17} + 61979011448 p^{2} T^{18} + 6984126676 p^{3} T^{19} - 283495188 p^{4} T^{20} - 170919005 p^{5} T^{21} + 39391395 p^{6} T^{22} + 8390856 p^{7} T^{23} - 115321 p^{8} T^{24} - 3617 p^{10} T^{25} - 2925 p^{10} T^{26} + 10289 p^{11} T^{27} + 654 p^{12} T^{28} - 100 p^{13} T^{29} - 7 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 31 | \( 1 + 2 T - 111 T^{2} - 689 T^{3} + 4450 T^{4} + 53717 T^{5} + 55596 T^{6} - 1624937 T^{7} - 10297837 T^{8} - 6998941 T^{9} + 304901363 T^{10} + 1814542788 T^{11} - 1032352178 T^{12} - 48418737280 T^{13} - 128226572438 T^{14} + 14427056690 p T^{15} + 146076545913 p T^{16} + 14427056690 p^{2} T^{17} - 128226572438 p^{2} T^{18} - 48418737280 p^{3} T^{19} - 1032352178 p^{4} T^{20} + 1814542788 p^{5} T^{21} + 304901363 p^{6} T^{22} - 6998941 p^{7} T^{23} - 10297837 p^{8} T^{24} - 1624937 p^{9} T^{25} + 55596 p^{10} T^{26} + 53717 p^{11} T^{27} + 4450 p^{12} T^{28} - 689 p^{13} T^{29} - 111 p^{14} T^{30} + 2 p^{15} T^{31} + p^{16} T^{32} \) | |
| 37 | \( 1 + 16 T + 89 T^{2} + 522 T^{3} + 4719 T^{4} + 17614 T^{5} + 87238 T^{6} + 816620 T^{7} - 2222836 T^{8} - 30347156 T^{9} - 32897727 T^{10} - 1513379442 T^{11} - 7898651454 T^{12} + 21410541190 T^{13} - 154796997102 T^{14} - 786476307222 T^{15} + 8974477206601 T^{16} - 786476307222 p T^{17} - 154796997102 p^{2} T^{18} + 21410541190 p^{3} T^{19} - 7898651454 p^{4} T^{20} - 1513379442 p^{5} T^{21} - 32897727 p^{6} T^{22} - 30347156 p^{7} T^{23} - 2222836 p^{8} T^{24} + 816620 p^{9} T^{25} + 87238 p^{10} T^{26} + 17614 p^{11} T^{27} + 4719 p^{12} T^{28} + 522 p^{13} T^{29} + 89 p^{14} T^{30} + 16 p^{15} T^{31} + p^{16} T^{32} \) | |
| 41 | \( 1 + 11 T - 133 T^{2} - 1691 T^{3} + 9296 T^{4} + 148841 T^{5} - 207544 T^{6} - 8131222 T^{7} - 15780480 T^{8} + 283357409 T^{9} + 1808729464 T^{10} - 4847227751 T^{11} - 93880111204 T^{12} - 41170156502 T^{13} + 3334902532040 T^{14} + 51994412549 p T^{15} - 119773790877992 T^{16} + 51994412549 p^{2} T^{17} + 3334902532040 p^{2} T^{18} - 41170156502 p^{3} T^{19} - 93880111204 p^{4} T^{20} - 4847227751 p^{5} T^{21} + 1808729464 p^{6} T^{22} + 283357409 p^{7} T^{23} - 15780480 p^{8} T^{24} - 8131222 p^{9} T^{25} - 207544 p^{10} T^{26} + 148841 p^{11} T^{27} + 9296 p^{12} T^{28} - 1691 p^{13} T^{29} - 133 p^{14} T^{30} + 11 p^{15} T^{31} + p^{16} T^{32} \) | |
| 43 | \( ( 1 + 8 T + 149 T^{2} + 453 T^{3} + 7657 T^{4} - 3325 T^{5} + 326776 T^{6} - 443368 T^{7} + 16565854 T^{8} - 443368 p T^{9} + 326776 p^{2} T^{10} - 3325 p^{3} T^{11} + 7657 p^{4} T^{12} + 453 p^{5} T^{13} + 149 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 47 | \( 1 + 10 T - 167 T^{2} - 2247 T^{3} + 8682 T^{4} + 221639 T^{5} + 269846 T^{6} - 248455 p T^{7} - 63472141 T^{8} + 229604893 T^{9} + 3897700707 T^{10} + 13010025112 T^{11} - 90951090948 T^{12} - 1162388838824 T^{13} - 3199374101152 T^{14} + 27341952204120 T^{15} + 324002209832685 T^{16} + 27341952204120 p T^{17} - 3199374101152 p^{2} T^{18} - 1162388838824 p^{3} T^{19} - 90951090948 p^{4} T^{20} + 13010025112 p^{5} T^{21} + 3897700707 p^{6} T^{22} + 229604893 p^{7} T^{23} - 63472141 p^{8} T^{24} - 248455 p^{10} T^{25} + 269846 p^{10} T^{26} + 221639 p^{11} T^{27} + 8682 p^{12} T^{28} - 2247 p^{13} T^{29} - 167 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 53 | \( 1 + 9 T + 89 T^{2} - 59 T^{3} + 5314 T^{4} + 13505 T^{5} + 239707 T^{6} - 2699487 T^{7} + 15661395 T^{8} + 86369761 T^{9} + 1995419569 T^{10} + 68832491 T^{11} + 89869603586 T^{12} + 247617022862 T^{13} + 4999881034984 T^{14} - 13024276641822 T^{15} + 75169753936991 T^{16} - 13024276641822 p T^{17} + 4999881034984 p^{2} T^{18} + 247617022862 p^{3} T^{19} + 89869603586 p^{4} T^{20} + 68832491 p^{5} T^{21} + 1995419569 p^{6} T^{22} + 86369761 p^{7} T^{23} + 15661395 p^{8} T^{24} - 2699487 p^{9} T^{25} + 239707 p^{10} T^{26} + 13505 p^{11} T^{27} + 5314 p^{12} T^{28} - 59 p^{13} T^{29} + 89 p^{14} T^{30} + 9 p^{15} T^{31} + p^{16} T^{32} \) | |
| 59 | \( 1 + 60 T + 1480 T^{2} + 18012 T^{3} + 82480 T^{4} - 467192 T^{5} - 6129506 T^{6} + 20926140 T^{7} + 853156511 T^{8} + 9033823003 T^{9} + 60005124904 T^{10} + 180425590422 T^{11} - 1210381401849 T^{12} - 13028713915837 T^{13} + 55314962283529 T^{14} + 2053768959197572 T^{15} + 21344481797350156 T^{16} + 2053768959197572 p T^{17} + 55314962283529 p^{2} T^{18} - 13028713915837 p^{3} T^{19} - 1210381401849 p^{4} T^{20} + 180425590422 p^{5} T^{21} + 60005124904 p^{6} T^{22} + 9033823003 p^{7} T^{23} + 853156511 p^{8} T^{24} + 20926140 p^{9} T^{25} - 6129506 p^{10} T^{26} - 467192 p^{11} T^{27} + 82480 p^{12} T^{28} + 18012 p^{13} T^{29} + 1480 p^{14} T^{30} + 60 p^{15} T^{31} + p^{16} T^{32} \) | |
| 61 | \( 1 - 30 T + 250 T^{2} + 144 T^{3} - 400 T^{4} - 168556 T^{5} + 1730486 T^{6} - 8066160 T^{7} - 41434559 T^{8} + 757404404 T^{9} + 3605866466 T^{10} - 83684380886 T^{11} + 216344808096 T^{12} + 2398178643684 T^{13} - 19368418756324 T^{14} - 173309871432556 T^{15} + 3352563706819921 T^{16} - 173309871432556 p T^{17} - 19368418756324 p^{2} T^{18} + 2398178643684 p^{3} T^{19} + 216344808096 p^{4} T^{20} - 83684380886 p^{5} T^{21} + 3605866466 p^{6} T^{22} + 757404404 p^{7} T^{23} - 41434559 p^{8} T^{24} - 8066160 p^{9} T^{25} + 1730486 p^{10} T^{26} - 168556 p^{11} T^{27} - 400 p^{12} T^{28} + 144 p^{13} T^{29} + 250 p^{14} T^{30} - 30 p^{15} T^{31} + p^{16} T^{32} \) | |
| 67 | \( ( 1 + 2 T + 270 T^{2} + 503 T^{3} + 39979 T^{4} + 83042 T^{5} + 4051165 T^{6} + 8333955 T^{7} + 308068550 T^{8} + 8333955 p T^{9} + 4051165 p^{2} T^{10} + 83042 p^{3} T^{11} + 39979 p^{4} T^{12} + 503 p^{5} T^{13} + 270 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 71 | \( 1 + 10 T - 29 T^{2} - 1530 T^{3} - 4335 T^{4} + 74310 T^{5} + 360480 T^{6} - 8005190 T^{7} - 45548210 T^{8} + 662986870 T^{9} + 5522651903 T^{10} - 45275572420 T^{11} - 650677884862 T^{12} + 1471681904640 T^{13} + 45800435450230 T^{14} + 53089235023420 T^{15} - 2401318073345415 T^{16} + 53089235023420 p T^{17} + 45800435450230 p^{2} T^{18} + 1471681904640 p^{3} T^{19} - 650677884862 p^{4} T^{20} - 45275572420 p^{5} T^{21} + 5522651903 p^{6} T^{22} + 662986870 p^{7} T^{23} - 45548210 p^{8} T^{24} - 8005190 p^{9} T^{25} + 360480 p^{10} T^{26} + 74310 p^{11} T^{27} - 4335 p^{12} T^{28} - 1530 p^{13} T^{29} - 29 p^{14} T^{30} + 10 p^{15} T^{31} + p^{16} T^{32} \) | |
| 73 | \( 1 - T - 50 T^{2} - 507 T^{3} + 7607 T^{4} + 21365 T^{5} + 138694 T^{6} - 4230307 T^{7} + 48546996 T^{8} - 133794525 T^{9} - 721368690 T^{10} - 26863217595 T^{11} + 477470092072 T^{12} - 244290523656 T^{13} + 8508140408490 T^{14} - 147898643641722 T^{15} + 1291482547231153 T^{16} - 147898643641722 p T^{17} + 8508140408490 p^{2} T^{18} - 244290523656 p^{3} T^{19} + 477470092072 p^{4} T^{20} - 26863217595 p^{5} T^{21} - 721368690 p^{6} T^{22} - 133794525 p^{7} T^{23} + 48546996 p^{8} T^{24} - 4230307 p^{9} T^{25} + 138694 p^{10} T^{26} + 21365 p^{11} T^{27} + 7607 p^{12} T^{28} - 507 p^{13} T^{29} - 50 p^{14} T^{30} - p^{15} T^{31} + p^{16} T^{32} \) | |
| 79 | \( 1 - 19 T - 128 T^{2} + 5590 T^{3} - 35211 T^{4} - 348100 T^{5} + 8147088 T^{6} - 62589588 T^{7} - 189000410 T^{8} + 9683071842 T^{9} - 89877613568 T^{10} + 43192902619 T^{11} + 7854845334960 T^{12} - 93411603601668 T^{13} + 336347402887446 T^{14} + 4849591076509872 T^{15} - 78943161844075351 T^{16} + 4849591076509872 p T^{17} + 336347402887446 p^{2} T^{18} - 93411603601668 p^{3} T^{19} + 7854845334960 p^{4} T^{20} + 43192902619 p^{5} T^{21} - 89877613568 p^{6} T^{22} + 9683071842 p^{7} T^{23} - 189000410 p^{8} T^{24} - 62589588 p^{9} T^{25} + 8147088 p^{10} T^{26} - 348100 p^{11} T^{27} - 35211 p^{12} T^{28} + 5590 p^{13} T^{29} - 128 p^{14} T^{30} - 19 p^{15} T^{31} + p^{16} T^{32} \) | |
| 83 | \( 1 - 64 T + 1842 T^{2} - 31344 T^{3} + 345978 T^{4} - 2598340 T^{5} + 16880318 T^{6} - 199694132 T^{7} + 3432126335 T^{8} - 47134056729 T^{9} + 485238710502 T^{10} - 4069061081284 T^{11} + 33149400013237 T^{12} - 316488329872661 T^{13} + 3399487978652475 T^{14} - 35447692006399282 T^{15} + 338480481117166088 T^{16} - 35447692006399282 p T^{17} + 3399487978652475 p^{2} T^{18} - 316488329872661 p^{3} T^{19} + 33149400013237 p^{4} T^{20} - 4069061081284 p^{5} T^{21} + 485238710502 p^{6} T^{22} - 47134056729 p^{7} T^{23} + 3432126335 p^{8} T^{24} - 199694132 p^{9} T^{25} + 16880318 p^{10} T^{26} - 2598340 p^{11} T^{27} + 345978 p^{12} T^{28} - 31344 p^{13} T^{29} + 1842 p^{14} T^{30} - 64 p^{15} T^{31} + p^{16} T^{32} \) | |
| 89 | \( ( 1 - 12 T + 435 T^{2} - 5180 T^{3} + 98390 T^{4} - 1059316 T^{5} + 14546512 T^{6} - 136934240 T^{7} + 1521322145 T^{8} - 136934240 p T^{9} + 14546512 p^{2} T^{10} - 1059316 p^{3} T^{11} + 98390 p^{4} T^{12} - 5180 p^{5} T^{13} + 435 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} )^{2} \) | |
| 97 | \( 1 + 3 T - 415 T^{2} - 2087 T^{3} + 63998 T^{4} + 741351 T^{5} - 2617790 T^{6} - 136595056 T^{7} - 613777870 T^{8} + 11063029847 T^{9} + 126851472554 T^{10} + 158741308403 T^{11} - 9875174287586 T^{12} - 101173447635612 T^{13} - 151300174361360 T^{14} + 5445806412551039 T^{15} + 82041604659406072 T^{16} + 5445806412551039 p T^{17} - 151300174361360 p^{2} T^{18} - 101173447635612 p^{3} T^{19} - 9875174287586 p^{4} T^{20} + 158741308403 p^{5} T^{21} + 126851472554 p^{6} T^{22} + 11063029847 p^{7} T^{23} - 613777870 p^{8} T^{24} - 136595056 p^{9} T^{25} - 2617790 p^{10} T^{26} + 741351 p^{11} T^{27} + 63998 p^{12} T^{28} - 2087 p^{13} T^{29} - 415 p^{14} T^{30} + 3 p^{15} T^{31} + p^{16} T^{32} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.98980563497984996142581005561, −2.96193613254245662558449623945, −2.95912922188507247504429871620, −2.90264815076706215973267923311, −2.88935146980087531527174472314, −2.73212306210564655950369259556, −2.50118607708049874370505311208, −2.46037062408470898813005175671, −2.18921705627951998226304091303, −2.13839265976743314472399681472, −1.97801967935142203907742920592, −1.95667929985179625517128684199, −1.86624588444213050249406635990, −1.85890763937251369479753381001, −1.75781972340813314395795195879, −1.63197582396578458511627416570, −1.60784780823344360041870802958, −1.54111789333844679433075121655, −1.13939369424324292749274827464, −1.01947629311583898879203580177, −0.78808669248002906598628828555, −0.76989902079850965376661640246, −0.69658551534924840307116105096, −0.35988407640562081069616683947, −0.18913102105388060178570814365, 0.18913102105388060178570814365, 0.35988407640562081069616683947, 0.69658551534924840307116105096, 0.76989902079850965376661640246, 0.78808669248002906598628828555, 1.01947629311583898879203580177, 1.13939369424324292749274827464, 1.54111789333844679433075121655, 1.60784780823344360041870802958, 1.63197582396578458511627416570, 1.75781972340813314395795195879, 1.85890763937251369479753381001, 1.86624588444213050249406635990, 1.95667929985179625517128684199, 1.97801967935142203907742920592, 2.13839265976743314472399681472, 2.18921705627951998226304091303, 2.46037062408470898813005175671, 2.50118607708049874370505311208, 2.73212306210564655950369259556, 2.88935146980087531527174472314, 2.90264815076706215973267923311, 2.95912922188507247504429871620, 2.96193613254245662558449623945, 2.98980563497984996142581005561
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.