Dirichlet series
| L(s) = 1 | − 14·3-s − 6·5-s − 7·7-s + 105·9-s + 11-s − 13-s + 84·15-s − 6·17-s + 19-s + 98·21-s + 10·23-s − 22·25-s − 560·27-s − 6·29-s − 5·31-s − 14·33-s + 42·35-s − 12·37-s + 14·39-s − 21·41-s + 8·43-s − 630·45-s + 18·47-s − 34·49-s + 84·51-s − 53-s − 6·55-s + ⋯ |
| L(s) = 1 | − 8.08·3-s − 2.68·5-s − 2.64·7-s + 35·9-s + 0.301·11-s − 0.277·13-s + 21.6·15-s − 1.45·17-s + 0.229·19-s + 21.3·21-s + 2.08·23-s − 4.39·25-s − 107.·27-s − 1.11·29-s − 0.898·31-s − 2.43·33-s + 7.09·35-s − 1.97·37-s + 2.24·39-s − 3.27·41-s + 1.21·43-s − 93.9·45-s + 2.62·47-s − 4.85·49-s + 11.7·51-s − 0.137·53-s − 0.809·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{14} \cdot 503^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{14} \cdot 503^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{28} \cdot 3^{14} \cdot 503^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.65063\times 10^{23}\) |
| Root analytic conductor: | \(6.94245\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(14\) |
| Selberg data: | \((28,\ 2^{28} \cdot 3^{14} \cdot 503^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T )^{14} \) | |
| 503 | \( ( 1 - T )^{14} \) | |
| good | 5 | \( 1 + 6 T + 58 T^{2} + 278 T^{3} + 1562 T^{4} + 2 p^{5} T^{5} + 26331 T^{6} + 3624 p^{2} T^{7} + 62782 p T^{8} + 947418 T^{9} + 2821173 T^{10} + 1512494 p T^{11} + 19808806 T^{12} + 47466067 T^{13} + 110699411 T^{14} + 47466067 p T^{15} + 19808806 p^{2} T^{16} + 1512494 p^{4} T^{17} + 2821173 p^{4} T^{18} + 947418 p^{5} T^{19} + 62782 p^{7} T^{20} + 3624 p^{9} T^{21} + 26331 p^{8} T^{22} + 2 p^{14} T^{23} + 1562 p^{10} T^{24} + 278 p^{11} T^{25} + 58 p^{12} T^{26} + 6 p^{13} T^{27} + p^{14} T^{28} \) |
| 7 | \( 1 + p T + 83 T^{2} + 460 T^{3} + 3137 T^{4} + 14373 T^{5} + 72788 T^{6} + 284365 T^{7} + 168160 p T^{8} + 4014686 T^{9} + 14276610 T^{10} + 6184881 p T^{11} + 136252341 T^{12} + 372092100 T^{13} + 1053965492 T^{14} + 372092100 p T^{15} + 136252341 p^{2} T^{16} + 6184881 p^{4} T^{17} + 14276610 p^{4} T^{18} + 4014686 p^{5} T^{19} + 168160 p^{7} T^{20} + 284365 p^{7} T^{21} + 72788 p^{8} T^{22} + 14373 p^{9} T^{23} + 3137 p^{10} T^{24} + 460 p^{11} T^{25} + 83 p^{12} T^{26} + p^{14} T^{27} + p^{14} T^{28} \) | |
| 11 | \( 1 - T + 111 T^{2} - 135 T^{3} + 6023 T^{4} - 8048 T^{5} + 212027 T^{6} - 292342 T^{7} + 5403338 T^{8} - 7361881 T^{9} + 105288081 T^{10} - 136977781 T^{11} + 1615950645 T^{12} - 1945732808 T^{13} + 19832017913 T^{14} - 1945732808 p T^{15} + 1615950645 p^{2} T^{16} - 136977781 p^{3} T^{17} + 105288081 p^{4} T^{18} - 7361881 p^{5} T^{19} + 5403338 p^{6} T^{20} - 292342 p^{7} T^{21} + 212027 p^{8} T^{22} - 8048 p^{9} T^{23} + 6023 p^{10} T^{24} - 135 p^{11} T^{25} + 111 p^{12} T^{26} - p^{13} T^{27} + p^{14} T^{28} \) | |
| 13 | \( 1 + T + 122 T^{2} + 112 T^{3} + 7274 T^{4} + 6282 T^{5} + 282377 T^{6} + 233640 T^{7} + 8007151 T^{8} + 6381741 T^{9} + 176012700 T^{10} + 133882094 T^{11} + 3100320477 T^{12} + 2201881653 T^{13} + 44518848131 T^{14} + 2201881653 p T^{15} + 3100320477 p^{2} T^{16} + 133882094 p^{3} T^{17} + 176012700 p^{4} T^{18} + 6381741 p^{5} T^{19} + 8007151 p^{6} T^{20} + 233640 p^{7} T^{21} + 282377 p^{8} T^{22} + 6282 p^{9} T^{23} + 7274 p^{10} T^{24} + 112 p^{11} T^{25} + 122 p^{12} T^{26} + p^{13} T^{27} + p^{14} T^{28} \) | |
| 17 | \( 1 + 6 T + 196 T^{2} + 1095 T^{3} + 18171 T^{4} + 94871 T^{5} + 1061309 T^{6} + 5176901 T^{7} + 43827969 T^{8} + 198828909 T^{9} + 1359235081 T^{10} + 5682927218 T^{11} + 32753076644 T^{12} + 124466281164 T^{13} + 624296368626 T^{14} + 124466281164 p T^{15} + 32753076644 p^{2} T^{16} + 5682927218 p^{3} T^{17} + 1359235081 p^{4} T^{18} + 198828909 p^{5} T^{19} + 43827969 p^{6} T^{20} + 5176901 p^{7} T^{21} + 1061309 p^{8} T^{22} + 94871 p^{9} T^{23} + 18171 p^{10} T^{24} + 1095 p^{11} T^{25} + 196 p^{12} T^{26} + 6 p^{13} T^{27} + p^{14} T^{28} \) | |
| 19 | \( 1 - T + 166 T^{2} - 21 T^{3} + 13038 T^{4} + 10350 T^{5} + 647630 T^{6} + 1159858 T^{7} + 23021057 T^{8} + 65256595 T^{9} + 634111607 T^{10} + 2393747074 T^{11} + 14482366836 T^{12} + 62244693079 T^{13} + 290239897357 T^{14} + 62244693079 p T^{15} + 14482366836 p^{2} T^{16} + 2393747074 p^{3} T^{17} + 634111607 p^{4} T^{18} + 65256595 p^{5} T^{19} + 23021057 p^{6} T^{20} + 1159858 p^{7} T^{21} + 647630 p^{8} T^{22} + 10350 p^{9} T^{23} + 13038 p^{10} T^{24} - 21 p^{11} T^{25} + 166 p^{12} T^{26} - p^{13} T^{27} + p^{14} T^{28} \) | |
| 23 | \( 1 - 10 T + 216 T^{2} - 1939 T^{3} + 23449 T^{4} - 186556 T^{5} + 1671187 T^{6} - 11795776 T^{7} + 86901151 T^{8} - 546862368 T^{9} + 151293508 p T^{10} - 19624403658 T^{11} + 110515924766 T^{12} - 560445205284 T^{13} + 2826170678691 T^{14} - 560445205284 p T^{15} + 110515924766 p^{2} T^{16} - 19624403658 p^{3} T^{17} + 151293508 p^{5} T^{18} - 546862368 p^{5} T^{19} + 86901151 p^{6} T^{20} - 11795776 p^{7} T^{21} + 1671187 p^{8} T^{22} - 186556 p^{9} T^{23} + 23449 p^{10} T^{24} - 1939 p^{11} T^{25} + 216 p^{12} T^{26} - 10 p^{13} T^{27} + p^{14} T^{28} \) | |
| 29 | \( 1 + 6 T + 227 T^{2} + 1229 T^{3} + 25746 T^{4} + 124551 T^{5} + 1937152 T^{6} + 8437387 T^{7} + 109285471 T^{8} + 433876095 T^{9} + 4943002742 T^{10} + 18056046552 T^{11} + 185509142815 T^{12} + 624907920941 T^{13} + 5856466452307 T^{14} + 624907920941 p T^{15} + 185509142815 p^{2} T^{16} + 18056046552 p^{3} T^{17} + 4943002742 p^{4} T^{18} + 433876095 p^{5} T^{19} + 109285471 p^{6} T^{20} + 8437387 p^{7} T^{21} + 1937152 p^{8} T^{22} + 124551 p^{9} T^{23} + 25746 p^{10} T^{24} + 1229 p^{11} T^{25} + 227 p^{12} T^{26} + 6 p^{13} T^{27} + p^{14} T^{28} \) | |
| 31 | \( 1 + 5 T + 261 T^{2} + 1099 T^{3} + 33795 T^{4} + 121395 T^{5} + 2893326 T^{6} + 8978864 T^{7} + 184068577 T^{8} + 500190054 T^{9} + 9241536839 T^{10} + 22312475042 T^{11} + 378425193101 T^{12} + 823683012922 T^{13} + 12849822369873 T^{14} + 823683012922 p T^{15} + 378425193101 p^{2} T^{16} + 22312475042 p^{3} T^{17} + 9241536839 p^{4} T^{18} + 500190054 p^{5} T^{19} + 184068577 p^{6} T^{20} + 8978864 p^{7} T^{21} + 2893326 p^{8} T^{22} + 121395 p^{9} T^{23} + 33795 p^{10} T^{24} + 1099 p^{11} T^{25} + 261 p^{12} T^{26} + 5 p^{13} T^{27} + p^{14} T^{28} \) | |
| 37 | \( 1 + 12 T + 324 T^{2} + 2770 T^{3} + 42753 T^{4} + 276775 T^{5} + 3235986 T^{6} + 16488388 T^{7} + 4577007 p T^{8} + 721009651 T^{9} + 7294547160 T^{10} + 28871221876 T^{11} + 296641242514 T^{12} + 1160563633924 T^{13} + 11462334151770 T^{14} + 1160563633924 p T^{15} + 296641242514 p^{2} T^{16} + 28871221876 p^{3} T^{17} + 7294547160 p^{4} T^{18} + 721009651 p^{5} T^{19} + 4577007 p^{7} T^{20} + 16488388 p^{7} T^{21} + 3235986 p^{8} T^{22} + 276775 p^{9} T^{23} + 42753 p^{10} T^{24} + 2770 p^{11} T^{25} + 324 p^{12} T^{26} + 12 p^{13} T^{27} + p^{14} T^{28} \) | |
| 41 | \( 1 + 21 T + 481 T^{2} + 6357 T^{3} + 2107 p T^{4} + 838884 T^{5} + 8353230 T^{6} + 61726005 T^{7} + 474602859 T^{8} + 2503687769 T^{9} + 14273222280 T^{10} + 30075360938 T^{11} + 41972232506 T^{12} - 2096259939142 T^{13} - 10495300292395 T^{14} - 2096259939142 p T^{15} + 41972232506 p^{2} T^{16} + 30075360938 p^{3} T^{17} + 14273222280 p^{4} T^{18} + 2503687769 p^{5} T^{19} + 474602859 p^{6} T^{20} + 61726005 p^{7} T^{21} + 8353230 p^{8} T^{22} + 838884 p^{9} T^{23} + 2107 p^{11} T^{24} + 6357 p^{11} T^{25} + 481 p^{12} T^{26} + 21 p^{13} T^{27} + p^{14} T^{28} \) | |
| 43 | \( 1 - 8 T + 354 T^{2} - 1938 T^{3} + 1320 p T^{4} - 219355 T^{5} + 5976154 T^{6} - 17262883 T^{7} + 486530298 T^{8} - 1127367194 T^{9} + 32370895769 T^{10} - 62776797506 T^{11} + 1788115244316 T^{12} - 3011573704262 T^{13} + 83329654210912 T^{14} - 3011573704262 p T^{15} + 1788115244316 p^{2} T^{16} - 62776797506 p^{3} T^{17} + 32370895769 p^{4} T^{18} - 1127367194 p^{5} T^{19} + 486530298 p^{6} T^{20} - 17262883 p^{7} T^{21} + 5976154 p^{8} T^{22} - 219355 p^{9} T^{23} + 1320 p^{11} T^{24} - 1938 p^{11} T^{25} + 354 p^{12} T^{26} - 8 p^{13} T^{27} + p^{14} T^{28} \) | |
| 47 | \( 1 - 18 T + 473 T^{2} - 5721 T^{3} + 88561 T^{4} - 838235 T^{5} + 9953843 T^{6} - 80091839 T^{7} + 801886449 T^{8} - 5698841202 T^{9} + 50405804023 T^{10} - 326828281946 T^{11} + 2678298763406 T^{12} - 16489842494004 T^{13} + 129893488380185 T^{14} - 16489842494004 p T^{15} + 2678298763406 p^{2} T^{16} - 326828281946 p^{3} T^{17} + 50405804023 p^{4} T^{18} - 5698841202 p^{5} T^{19} + 801886449 p^{6} T^{20} - 80091839 p^{7} T^{21} + 9953843 p^{8} T^{22} - 838235 p^{9} T^{23} + 88561 p^{10} T^{24} - 5721 p^{11} T^{25} + 473 p^{12} T^{26} - 18 p^{13} T^{27} + p^{14} T^{28} \) | |
| 53 | \( 1 + T + 390 T^{2} - 262 T^{3} + 75166 T^{4} - 163394 T^{5} + 9790723 T^{6} - 32589030 T^{7} + 983567676 T^{8} - 3933196150 T^{9} + 1526037788 p T^{10} - 339939296676 T^{11} + 5561314668583 T^{12} - 22658964842730 T^{13} + 321398332607647 T^{14} - 22658964842730 p T^{15} + 5561314668583 p^{2} T^{16} - 339939296676 p^{3} T^{17} + 1526037788 p^{5} T^{18} - 3933196150 p^{5} T^{19} + 983567676 p^{6} T^{20} - 32589030 p^{7} T^{21} + 9790723 p^{8} T^{22} - 163394 p^{9} T^{23} + 75166 p^{10} T^{24} - 262 p^{11} T^{25} + 390 p^{12} T^{26} + p^{13} T^{27} + p^{14} T^{28} \) | |
| 59 | \( 1 - 14 T + 452 T^{2} - 4322 T^{3} + 78638 T^{4} - 521049 T^{5} + 6923590 T^{6} - 28320427 T^{7} + 310018991 T^{8} - 344454210 T^{9} + 4093714768 T^{10} + 29171121678 T^{11} - 250440926296 T^{12} + 967433674354 T^{13} - 19218817079819 T^{14} + 967433674354 p T^{15} - 250440926296 p^{2} T^{16} + 29171121678 p^{3} T^{17} + 4093714768 p^{4} T^{18} - 344454210 p^{5} T^{19} + 310018991 p^{6} T^{20} - 28320427 p^{7} T^{21} + 6923590 p^{8} T^{22} - 521049 p^{9} T^{23} + 78638 p^{10} T^{24} - 4322 p^{11} T^{25} + 452 p^{12} T^{26} - 14 p^{13} T^{27} + p^{14} T^{28} \) | |
| 61 | \( 1 + 19 T + 583 T^{2} + 8507 T^{3} + 2564 p T^{4} + 1889437 T^{5} + 26505968 T^{6} + 276232681 T^{7} + 3238426200 T^{8} + 29919430179 T^{9} + 307821241368 T^{10} + 2574193615139 T^{11} + 23964991029635 T^{12} + 184198427484908 T^{13} + 1579458247956783 T^{14} + 184198427484908 p T^{15} + 23964991029635 p^{2} T^{16} + 2574193615139 p^{3} T^{17} + 307821241368 p^{4} T^{18} + 29919430179 p^{5} T^{19} + 3238426200 p^{6} T^{20} + 276232681 p^{7} T^{21} + 26505968 p^{8} T^{22} + 1889437 p^{9} T^{23} + 2564 p^{11} T^{24} + 8507 p^{11} T^{25} + 583 p^{12} T^{26} + 19 p^{13} T^{27} + p^{14} T^{28} \) | |
| 67 | \( 1 - 17 T + 743 T^{2} - 10871 T^{3} + 265080 T^{4} - 3383136 T^{5} + 60227118 T^{6} - 678275198 T^{7} + 9745973268 T^{8} - 97652114071 T^{9} + 1189453063149 T^{10} - 10651817747195 T^{11} + 112990683237438 T^{12} - 905106476523372 T^{13} + 8490493017296618 T^{14} - 905106476523372 p T^{15} + 112990683237438 p^{2} T^{16} - 10651817747195 p^{3} T^{17} + 1189453063149 p^{4} T^{18} - 97652114071 p^{5} T^{19} + 9745973268 p^{6} T^{20} - 678275198 p^{7} T^{21} + 60227118 p^{8} T^{22} - 3383136 p^{9} T^{23} + 265080 p^{10} T^{24} - 10871 p^{11} T^{25} + 743 p^{12} T^{26} - 17 p^{13} T^{27} + p^{14} T^{28} \) | |
| 71 | \( 1 + 13 T + 414 T^{2} + 3380 T^{3} + 68701 T^{4} + 375534 T^{5} + 7416577 T^{6} + 31234087 T^{7} + 725206675 T^{8} + 2896681910 T^{9} + 67031322512 T^{10} + 237112194174 T^{11} + 5222205503552 T^{12} + 14827269124340 T^{13} + 368025813733024 T^{14} + 14827269124340 p T^{15} + 5222205503552 p^{2} T^{16} + 237112194174 p^{3} T^{17} + 67031322512 p^{4} T^{18} + 2896681910 p^{5} T^{19} + 725206675 p^{6} T^{20} + 31234087 p^{7} T^{21} + 7416577 p^{8} T^{22} + 375534 p^{9} T^{23} + 68701 p^{10} T^{24} + 3380 p^{11} T^{25} + 414 p^{12} T^{26} + 13 p^{13} T^{27} + p^{14} T^{28} \) | |
| 73 | \( 1 + 12 T + 541 T^{2} + 5015 T^{3} + 1908 p T^{4} + 1030216 T^{5} + 23298209 T^{6} + 138310964 T^{7} + 2899077033 T^{8} + 13784429115 T^{9} + 291536859944 T^{10} + 1125711264561 T^{11} + 25142453455456 T^{12} + 83468872411010 T^{13} + 1931710095428119 T^{14} + 83468872411010 p T^{15} + 25142453455456 p^{2} T^{16} + 1125711264561 p^{3} T^{17} + 291536859944 p^{4} T^{18} + 13784429115 p^{5} T^{19} + 2899077033 p^{6} T^{20} + 138310964 p^{7} T^{21} + 23298209 p^{8} T^{22} + 1030216 p^{9} T^{23} + 1908 p^{11} T^{24} + 5015 p^{11} T^{25} + 541 p^{12} T^{26} + 12 p^{13} T^{27} + p^{14} T^{28} \) | |
| 79 | \( 1 + 8 T + 615 T^{2} + 3845 T^{3} + 180030 T^{4} + 873585 T^{5} + 34014811 T^{6} + 124470739 T^{7} + 4745819278 T^{8} + 12494836529 T^{9} + 530294511516 T^{10} + 964343335912 T^{11} + 50231604288570 T^{12} + 66951805620120 T^{13} + 4194826354383810 T^{14} + 66951805620120 p T^{15} + 50231604288570 p^{2} T^{16} + 964343335912 p^{3} T^{17} + 530294511516 p^{4} T^{18} + 12494836529 p^{5} T^{19} + 4745819278 p^{6} T^{20} + 124470739 p^{7} T^{21} + 34014811 p^{8} T^{22} + 873585 p^{9} T^{23} + 180030 p^{10} T^{24} + 3845 p^{11} T^{25} + 615 p^{12} T^{26} + 8 p^{13} T^{27} + p^{14} T^{28} \) | |
| 83 | \( 1 - 11 T + 656 T^{2} - 5056 T^{3} + 192317 T^{4} - 959632 T^{5} + 34657195 T^{6} - 87214970 T^{7} + 4558328492 T^{8} - 1765125372 T^{9} + 501708447367 T^{10} + 494206905927 T^{11} + 49776214226460 T^{12} + 72247657065423 T^{13} + 4414188170499223 T^{14} + 72247657065423 p T^{15} + 49776214226460 p^{2} T^{16} + 494206905927 p^{3} T^{17} + 501708447367 p^{4} T^{18} - 1765125372 p^{5} T^{19} + 4558328492 p^{6} T^{20} - 87214970 p^{7} T^{21} + 34657195 p^{8} T^{22} - 959632 p^{9} T^{23} + 192317 p^{10} T^{24} - 5056 p^{11} T^{25} + 656 p^{12} T^{26} - 11 p^{13} T^{27} + p^{14} T^{28} \) | |
| 89 | \( 1 + 9 T + 719 T^{2} + 5654 T^{3} + 250799 T^{4} + 1744954 T^{5} + 57326462 T^{6} + 355900040 T^{7} + 9754216470 T^{8} + 54483622147 T^{9} + 1325899407645 T^{10} + 6727184441231 T^{11} + 150016815772611 T^{12} + 698396014652661 T^{13} + 14432101932699162 T^{14} + 698396014652661 p T^{15} + 150016815772611 p^{2} T^{16} + 6727184441231 p^{3} T^{17} + 1325899407645 p^{4} T^{18} + 54483622147 p^{5} T^{19} + 9754216470 p^{6} T^{20} + 355900040 p^{7} T^{21} + 57326462 p^{8} T^{22} + 1744954 p^{9} T^{23} + 250799 p^{10} T^{24} + 5654 p^{11} T^{25} + 719 p^{12} T^{26} + 9 p^{13} T^{27} + p^{14} T^{28} \) | |
| 97 | \( 1 + 25 T + 794 T^{2} + 14844 T^{3} + 289128 T^{4} + 4261325 T^{5} + 63287826 T^{6} + 758679247 T^{7} + 9154798945 T^{8} + 90406995183 T^{9} + 916189532616 T^{10} + 7540054718201 T^{11} + 69062971714944 T^{12} + 524234361513402 T^{13} + 5466935127344123 T^{14} + 524234361513402 p T^{15} + 69062971714944 p^{2} T^{16} + 7540054718201 p^{3} T^{17} + 916189532616 p^{4} T^{18} + 90406995183 p^{5} T^{19} + 9154798945 p^{6} T^{20} + 758679247 p^{7} T^{21} + 63287826 p^{8} T^{22} + 4261325 p^{9} T^{23} + 289128 p^{10} T^{24} + 14844 p^{11} T^{25} + 794 p^{12} T^{26} + 25 p^{13} T^{27} + p^{14} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.42387875235785624661632302287, −2.38110371191983144082406179743, −2.37548269055608191238043076118, −2.26491315669605817756529673953, −2.21606274282366568081043890306, −2.20266996441174951209809457193, −2.13808382112841555143996514893, −2.08025348247524219774908865228, −2.06735111370792006633093516879, −2.06259165395853570015288621548, −2.05641065732382314450949698671, −1.43738588709128446243121710751, −1.43635318635286779898899064374, −1.41638972353604963963656454053, −1.41245618299848068384229730977, −1.35462665182713482784805941562, −1.34695053114204813432586721143, −1.33377270134268214652427432601, −1.16194482853384384930765018480, −1.15100260388262728991956245665, −1.10901471719398478190242607592, −1.06746804396542010812999897419, −0.999912533148164814218315372149, −0.922181670521683617191921464595, −0.827633521603719245143926553171, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.827633521603719245143926553171, 0.922181670521683617191921464595, 0.999912533148164814218315372149, 1.06746804396542010812999897419, 1.10901471719398478190242607592, 1.15100260388262728991956245665, 1.16194482853384384930765018480, 1.33377270134268214652427432601, 1.34695053114204813432586721143, 1.35462665182713482784805941562, 1.41245618299848068384229730977, 1.41638972353604963963656454053, 1.43635318635286779898899064374, 1.43738588709128446243121710751, 2.05641065732382314450949698671, 2.06259165395853570015288621548, 2.06735111370792006633093516879, 2.08025348247524219774908865228, 2.13808382112841555143996514893, 2.20266996441174951209809457193, 2.21606274282366568081043890306, 2.26491315669605817756529673953, 2.37548269055608191238043076118, 2.38110371191983144082406179743, 2.42387875235785624661632302287
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.