L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.984 + 0.173i)3-s + (−0.939 + 0.342i)4-s − i·6-s + (−0.642 + 0.766i)7-s + (0.5 + 0.866i)8-s + (0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.984 + 0.173i)12-s + (0.939 − 0.342i)13-s + (0.866 + 0.5i)14-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + (0.173 − 0.984i)18-s + (0.984 + 0.173i)19-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.984 + 0.173i)3-s + (−0.939 + 0.342i)4-s − i·6-s + (−0.642 + 0.766i)7-s + (0.5 + 0.866i)8-s + (0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.984 + 0.173i)12-s + (0.939 − 0.342i)13-s + (0.866 + 0.5i)14-s + (0.766 − 0.642i)16-s + (−0.939 − 0.342i)17-s + (0.173 − 0.984i)18-s + (0.984 + 0.173i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.292771010 - 0.2608155546i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.292771010 - 0.2608155546i\) |
\(L(1)\) |
\(\approx\) |
\(1.169767740 - 0.2797809041i\) |
\(L(1)\) |
\(\approx\) |
\(1.169767740 - 0.2797809041i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.984 + 0.173i)T \) |
| 7 | \( 1 + (-0.642 + 0.766i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.939 - 0.342i)T \) |
| 17 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 + (0.984 + 0.173i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.866 - 0.5i)T \) |
| 53 | \( 1 + (-0.642 - 0.766i)T \) |
| 59 | \( 1 + (0.642 + 0.766i)T \) |
| 61 | \( 1 + (-0.342 - 0.939i)T \) |
| 67 | \( 1 + (0.642 - 0.766i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (-0.642 + 0.766i)T \) |
| 83 | \( 1 + (-0.342 + 0.939i)T \) |
| 89 | \( 1 + (-0.642 - 0.766i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.66289135239292804526741099478, −26.40338468732006463266693868652, −25.44489480503471499295297637691, −24.49083407574098021280956097953, −23.810833501573735801111889403580, −22.69995281381770806917310928171, −21.64724034987770493547097925875, −20.31959746450326100819179346317, −19.37330950784525571220915372212, −18.673115143316727919023564195855, −17.48086179579090097802479475353, −16.34612214384411381841800321475, −15.62970845659695839432062514245, −14.50290087474273664794723316375, −13.45230791938976006424558452950, −13.24550703394424693238836511688, −11.1747268473275288224826045133, −9.698885157286823108161091119199, −9.01859109272841852660842936302, −7.94096221906438147256409358389, −6.949160153843633359353097462269, −6.020997740884327613813015713584, −4.20859216040772414370913143647, −3.38027543278947848188989659489, −1.218254753931942635375160543923,
1.6599814210656529644133078572, 2.82980485842328893704212787375, 3.75723602864349514393125869740, 5.04876798968879493637354056553, 6.896876943337655717330449247249, 8.37023123653320073122102577730, 9.153110165818843425738209495379, 9.89392686242685956255308587852, 11.13353689978203604884312126835, 12.43928702879311465331725153710, 13.133275085598277437244511854340, 14.20639891643686330002176740642, 15.28462970729422828891013076726, 16.33974163314060600628681871280, 17.98592593602903029596470703338, 18.57312317502621198969656874831, 19.70517649799954751794119171241, 20.2771943778509906584216397142, 21.17731367296214493463796437436, 22.249859105394112451479489835221, 22.87095490214998200229500897510, 24.580022697468005066200479348491, 25.47599559473309929190405838785, 26.26391280530288826777890630243, 27.22344102963871176659595098043