L(s) = 1 | + 1.73·5-s + (−1 − 2.44i)7-s − 4.24i·11-s + 2.44i·13-s + 1.73·17-s − 2.44i·19-s − 8.48i·23-s − 2.00·25-s − 4.24i·29-s + 7.34i·31-s + (−1.73 − 4.24i)35-s + 37-s − 1.73·41-s + 7·43-s + 12.1·47-s + ⋯ |
L(s) = 1 | + 0.774·5-s + (−0.377 − 0.925i)7-s − 1.27i·11-s + 0.679i·13-s + 0.420·17-s − 0.561i·19-s − 1.76i·23-s − 0.400·25-s − 0.787i·29-s + 1.31i·31-s + (−0.292 − 0.717i)35-s + 0.164·37-s − 0.270·41-s + 1.06·43-s + 1.76·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.377 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.30026 - 0.873614i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.30026 - 0.873614i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1 + 2.44i)T \) |
good | 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 2.44iT - 13T^{2} \) |
| 17 | \( 1 - 1.73T + 17T^{2} \) |
| 19 | \( 1 + 2.44iT - 19T^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 + 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 7.34iT - 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 1.73T + 41T^{2} \) |
| 43 | \( 1 - 7T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 8.66T + 59T^{2} \) |
| 61 | \( 1 + 2.44iT - 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 8.48iT - 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 5T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 2.44iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.34426525859597704463551094021, −9.307913880278323664256075392099, −8.627052354356705930261600780148, −7.52686071648128751915038652967, −6.53208048026689920608092470947, −5.94167479771536022367367964805, −4.70091072826839444226978786153, −3.64766738488151746817079886887, −2.45577270716896707289794138655, −0.825239729247605122427609875143,
1.71418341039061882413093855062, 2.74665393252395637760540054506, 4.05125065144832968593126507141, 5.56490975815474180404195894810, 5.70997031704695878768360658482, 7.07667180901742417189476096528, 7.85769629247980859625780889293, 9.029108492514994409750931806313, 9.721644776378650718797070189679, 10.17610380946572966534954011485