| L(s) = 1 | + 2-s + 4-s − 3·5-s − 5·7-s + 8-s − 3·10-s + 4·11-s − 6·13-s − 5·14-s + 16-s + 17-s − 5·19-s − 3·20-s + 4·22-s − 6·23-s + 4·25-s − 6·26-s − 5·28-s + 29-s + 32-s + 34-s + 15·35-s + 37-s − 5·38-s − 3·40-s − 7·41-s + 43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s − 1.88·7-s + 0.353·8-s − 0.948·10-s + 1.20·11-s − 1.66·13-s − 1.33·14-s + 1/4·16-s + 0.242·17-s − 1.14·19-s − 0.670·20-s + 0.852·22-s − 1.25·23-s + 4/5·25-s − 1.17·26-s − 0.944·28-s + 0.185·29-s + 0.176·32-s + 0.171·34-s + 2.53·35-s + 0.164·37-s − 0.811·38-s − 0.474·40-s − 1.09·41-s + 0.152·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−10.39346098329972959714317376635, −9.697096358908895094097979196848, −8.624040339242858735116607625652, −7.35873627125951208034776293560, −6.80458191599030092067206255244, −5.86302600758819880324853381601, −4.29436665212333997528993193552, −3.77198990878818776634290664501, −2.63662744533589973731437635315, 0,
2.63662744533589973731437635315, 3.77198990878818776634290664501, 4.29436665212333997528993193552, 5.86302600758819880324853381601, 6.80458191599030092067206255244, 7.35873627125951208034776293560, 8.624040339242858735116607625652, 9.697096358908895094097979196848, 10.39346098329972959714317376635
