| L(s) = 1 | + 2.85·3-s − 5-s − 0.624·7-s + 5.13·9-s − 4.95·11-s − 3.40·13-s − 2.85·15-s − 6.87·17-s − 4.33·19-s − 1.78·21-s + 5.91·23-s + 25-s + 6.08·27-s − 0.438·29-s + 9.05·31-s − 14.1·33-s + 0.624·35-s − 37-s − 9.70·39-s + 7.16·41-s − 9.12·43-s − 5.13·45-s − 11.6·47-s − 6.60·49-s − 19.6·51-s + 5.85·53-s + 4.95·55-s + ⋯ |
| L(s) = 1 | + 1.64·3-s − 0.447·5-s − 0.236·7-s + 1.71·9-s − 1.49·11-s − 0.943·13-s − 0.736·15-s − 1.66·17-s − 0.993·19-s − 0.388·21-s + 1.23·23-s + 0.200·25-s + 1.17·27-s − 0.0813·29-s + 1.62·31-s − 2.46·33-s + 0.105·35-s − 0.164·37-s − 1.55·39-s + 1.11·41-s − 1.39·43-s − 0.765·45-s − 1.70·47-s − 0.944·49-s − 2.74·51-s + 0.804·53-s + 0.668·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 2.85T + 3T^{2} \) |
| 7 | \( 1 + 0.624T + 7T^{2} \) |
| 11 | \( 1 + 4.95T + 11T^{2} \) |
| 13 | \( 1 + 3.40T + 13T^{2} \) |
| 17 | \( 1 + 6.87T + 17T^{2} \) |
| 19 | \( 1 + 4.33T + 19T^{2} \) |
| 23 | \( 1 - 5.91T + 23T^{2} \) |
| 29 | \( 1 + 0.438T + 29T^{2} \) |
| 31 | \( 1 - 9.05T + 31T^{2} \) |
| 41 | \( 1 - 7.16T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 5.85T + 53T^{2} \) |
| 59 | \( 1 + 15.2T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 - 5.53T + 71T^{2} \) |
| 73 | \( 1 - 7.53T + 73T^{2} \) |
| 79 | \( 1 + 1.98T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 - 9.84T + 89T^{2} \) |
| 97 | \( 1 + 0.874T + 97T^{2} \) |
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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
−8.258905731746397818792480111992, −7.88665171455590010235164185642, −7.07994051385097410117499902828, −6.36589675647046854032727259242, −4.76881875936953142116081098891, −4.55948540468960895003602892796, −3.21369730652761877350671077938, −2.73502122158421810573997979791, −1.95576556454776747627436466183, 0,
1.95576556454776747627436466183, 2.73502122158421810573997979791, 3.21369730652761877350671077938, 4.55948540468960895003602892796, 4.76881875936953142116081098891, 6.36589675647046854032727259242, 7.07994051385097410117499902828, 7.88665171455590010235164185642, 8.258905731746397818792480111992
