| L(s) = 1 | + 2-s + 2.79·3-s + 4-s + 0.290·5-s + 2.79·6-s + 0.316·7-s + 8-s + 4.79·9-s + 0.290·10-s + 11-s + 2.79·12-s + 3.48·13-s + 0.316·14-s + 0.812·15-s + 16-s − 3.21·17-s + 4.79·18-s − 4.71·19-s + 0.290·20-s + 0.883·21-s + 22-s − 0.819·23-s + 2.79·24-s − 4.91·25-s + 3.48·26-s + 5.00·27-s + 0.316·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.61·3-s + 0.5·4-s + 0.130·5-s + 1.13·6-s + 0.119·7-s + 0.353·8-s + 1.59·9-s + 0.0919·10-s + 0.301·11-s + 0.805·12-s + 0.965·13-s + 0.0845·14-s + 0.209·15-s + 0.250·16-s − 0.780·17-s + 1.12·18-s − 1.08·19-s + 0.0650·20-s + 0.192·21-s + 0.213·22-s − 0.170·23-s + 0.569·24-s − 0.983·25-s + 0.682·26-s + 0.963·27-s + 0.0597·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.450399461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.450399461\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| good | 3 | \( 1 - 2.79T + 3T^{2} \) |
| 5 | \( 1 - 0.290T + 5T^{2} \) |
| 7 | \( 1 - 0.316T + 7T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 17 | \( 1 + 3.21T + 17T^{2} \) |
| 19 | \( 1 + 4.71T + 19T^{2} \) |
| 23 | \( 1 + 0.819T + 23T^{2} \) |
| 29 | \( 1 + 3.32T + 29T^{2} \) |
| 31 | \( 1 + 8.55T + 31T^{2} \) |
| 37 | \( 1 + 7.45T + 37T^{2} \) |
| 41 | \( 1 - 7.16T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 11.5T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 5.51T + 71T^{2} \) |
| 73 | \( 1 - 16.1T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 7.02T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−9.485360504542138795779976299941, −8.957673025265923178872045764521, −8.177155575233862539175677292063, −7.39831677617733301195102081193, −6.48353346547178987354982347286, −5.51531044735401931940213485279, −4.00045437663005137406587618780, −3.86150205845430117685185163966, −2.50774892424078626704790376812, −1.77873609165437473619620730807,
1.77873609165437473619620730807, 2.50774892424078626704790376812, 3.86150205845430117685185163966, 4.00045437663005137406587618780, 5.51531044735401931940213485279, 6.48353346547178987354982347286, 7.39831677617733301195102081193, 8.177155575233862539175677292063, 8.957673025265923178872045764521, 9.485360504542138795779976299941
