| L(s) = 1 | + 2-s − 3-s + 4-s − 1.03·5-s − 6-s + 7-s + 8-s + 9-s − 1.03·10-s − 0.0170·11-s − 12-s − 0.779·13-s + 14-s + 1.03·15-s + 16-s − 1.49·17-s + 18-s − 4.68·19-s − 1.03·20-s − 21-s − 0.0170·22-s + 8.09·23-s − 24-s − 3.92·25-s − 0.779·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.464·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.328·10-s − 0.00514·11-s − 0.288·12-s − 0.216·13-s + 0.267·14-s + 0.267·15-s + 0.250·16-s − 0.363·17-s + 0.235·18-s − 1.07·19-s − 0.232·20-s − 0.218·21-s − 0.00363·22-s + 1.68·23-s − 0.204·24-s − 0.784·25-s − 0.152·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 191 | \( 1 + T \) |
| good | 5 | \( 1 + 1.03T + 5T^{2} \) |
| 11 | \( 1 + 0.0170T + 11T^{2} \) |
| 13 | \( 1 + 0.779T + 13T^{2} \) |
| 17 | \( 1 + 1.49T + 17T^{2} \) |
| 19 | \( 1 + 4.68T + 19T^{2} \) |
| 23 | \( 1 - 8.09T + 23T^{2} \) |
| 29 | \( 1 - 2.59T + 29T^{2} \) |
| 31 | \( 1 + 9.31T + 31T^{2} \) |
| 37 | \( 1 - 1.14T + 37T^{2} \) |
| 41 | \( 1 + 3.57T + 41T^{2} \) |
| 43 | \( 1 - 11.7T + 43T^{2} \) |
| 47 | \( 1 - 9.74T + 47T^{2} \) |
| 53 | \( 1 + 7.31T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 + 0.973T + 67T^{2} \) |
| 71 | \( 1 + 8.70T + 71T^{2} \) |
| 73 | \( 1 + 5.14T + 73T^{2} \) |
| 79 | \( 1 + 2.68T + 79T^{2} \) |
| 83 | \( 1 - 2.20T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 - 1.17T + 97T^{2} \) |
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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
−7.41157817766617217467572824556, −6.72668083857862752996289932994, −5.99877840166505484915391522821, −5.36992091765656834260338919302, −4.56522692690655586943397921463, −4.17482090612832788080108492792, −3.22089351578375831970196705952, −2.30771548943943154802646492838, −1.33312670309916574175423791483, 0,
1.33312670309916574175423791483, 2.30771548943943154802646492838, 3.22089351578375831970196705952, 4.17482090612832788080108492792, 4.56522692690655586943397921463, 5.36992091765656834260338919302, 5.99877840166505484915391522821, 6.72668083857862752996289932994, 7.41157817766617217467572824556
