| L(s) = 1 | + 0.0694·2-s + 2.28·3-s − 1.99·4-s + 0.158·6-s − 0.976·7-s − 0.277·8-s + 2.22·9-s + 0.634·11-s − 4.56·12-s + 1.72·13-s − 0.0677·14-s + 3.97·16-s + 3.57·17-s + 0.154·18-s − 8.41·19-s − 2.23·21-s + 0.0440·22-s − 3.47·23-s − 0.634·24-s + 0.119·26-s − 1.76·27-s + 1.94·28-s − 5.33·29-s + 2.47·31-s + 0.830·32-s + 1.45·33-s + 0.248·34-s + ⋯ |
| L(s) = 1 | + 0.0491·2-s + 1.32·3-s − 0.997·4-s + 0.0648·6-s − 0.368·7-s − 0.0980·8-s + 0.743·9-s + 0.191·11-s − 1.31·12-s + 0.477·13-s − 0.0181·14-s + 0.992·16-s + 0.866·17-s + 0.0364·18-s − 1.93·19-s − 0.487·21-s + 0.00939·22-s − 0.724·23-s − 0.129·24-s + 0.0234·26-s − 0.339·27-s + 0.368·28-s − 0.991·29-s + 0.444·31-s + 0.146·32-s + 0.252·33-s + 0.0425·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 - 0.0694T + 2T^{2} \) |
| 3 | \( 1 - 2.28T + 3T^{2} \) |
| 7 | \( 1 + 0.976T + 7T^{2} \) |
| 11 | \( 1 - 0.634T + 11T^{2} \) |
| 13 | \( 1 - 1.72T + 13T^{2} \) |
| 17 | \( 1 - 3.57T + 17T^{2} \) |
| 19 | \( 1 + 8.41T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 + 5.33T + 29T^{2} \) |
| 31 | \( 1 - 2.47T + 31T^{2} \) |
| 37 | \( 1 - 3.69T + 37T^{2} \) |
| 41 | \( 1 - 8.97T + 41T^{2} \) |
| 43 | \( 1 + 9.17T + 43T^{2} \) |
| 47 | \( 1 - 5.13T + 47T^{2} \) |
| 53 | \( 1 + 7.54T + 53T^{2} \) |
| 59 | \( 1 - 14.0T + 59T^{2} \) |
| 61 | \( 1 - 8.35T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 6.10T + 71T^{2} \) |
| 73 | \( 1 + 9.09T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 14.3T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 1.27T + 97T^{2} \) |
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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
−8.009391042516638001464618823036, −7.25529700407799465316822207773, −6.18645598248752817813581047122, −5.67045859534367842864569675240, −4.47011624785944640555651512200, −3.97694142357188443437953543981, −3.33413851722279296843653187806, −2.47538312142213154834691803117, −1.45540353713700557832812531100, 0,
1.45540353713700557832812531100, 2.47538312142213154834691803117, 3.33413851722279296843653187806, 3.97694142357188443437953543981, 4.47011624785944640555651512200, 5.67045859534367842864569675240, 6.18645598248752817813581047122, 7.25529700407799465316822207773, 8.009391042516638001464618823036
