| L(s) = 1 | + 3.82·2-s + 16.7·3-s − 17.3·4-s − 25·5-s + 64.2·6-s + 68.9·7-s − 188.·8-s + 39.1·9-s − 95.5·10-s + 121·11-s − 292.·12-s − 1.03e3·13-s + 263.·14-s − 419.·15-s − 164.·16-s − 891.·17-s + 149.·18-s − 361·19-s + 434.·20-s + 1.15e3·21-s + 462.·22-s + 3.27e3·23-s − 3.17e3·24-s + 625·25-s − 3.95e3·26-s − 3.42e3·27-s − 1.19e3·28-s + ⋯ |
| L(s) = 1 | + 0.675·2-s + 1.07·3-s − 0.543·4-s − 0.447·5-s + 0.728·6-s + 0.531·7-s − 1.04·8-s + 0.161·9-s − 0.302·10-s + 0.301·11-s − 0.585·12-s − 1.69·13-s + 0.359·14-s − 0.481·15-s − 0.160·16-s − 0.748·17-s + 0.108·18-s − 0.229·19-s + 0.243·20-s + 0.573·21-s + 0.203·22-s + 1.29·23-s − 1.12·24-s + 0.200·25-s − 1.14·26-s − 0.903·27-s − 0.289·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.750984014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.750984014\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 3.82T + 32T^{2} \) |
| 3 | \( 1 - 16.7T + 243T^{2} \) |
| 7 | \( 1 - 68.9T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.03e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 891.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.27e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.31e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.46e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.27e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.42e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.69e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.83e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.22e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.84e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.56e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.77e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.98e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.65e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.21e5T + 8.58e9T^{2} \) |
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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
−9.322534488729521985836328647578, −8.312903359462835798854023328937, −7.72434306844614590675020528774, −6.75255305076771012273219262750, −5.47064241478269929425652549314, −4.64713654561994208030071331522, −3.99729567449675946252131193737, −2.93761755407244755314059013565, −2.25231066488851849755809224482, −0.59092609829095570892424182277,
0.59092609829095570892424182277, 2.25231066488851849755809224482, 2.93761755407244755314059013565, 3.99729567449675946252131193737, 4.64713654561994208030071331522, 5.47064241478269929425652549314, 6.75255305076771012273219262750, 7.72434306844614590675020528774, 8.312903359462835798854023328937, 9.322534488729521985836328647578
