| L(s) = 1 | + 1.09·2-s − 3-s − 0.808·4-s + 5-s − 1.09·6-s + 0.212·7-s − 3.06·8-s + 9-s + 1.09·10-s + 3.48·11-s + 0.808·12-s − 3.29·13-s + 0.231·14-s − 15-s − 1.72·16-s − 0.372·17-s + 1.09·18-s − 3.68·19-s − 0.808·20-s − 0.212·21-s + 3.80·22-s + 3.62·23-s + 3.06·24-s + 25-s − 3.59·26-s − 27-s − 0.171·28-s + ⋯ |
| L(s) = 1 | + 0.771·2-s − 0.577·3-s − 0.404·4-s + 0.447·5-s − 0.445·6-s + 0.0802·7-s − 1.08·8-s + 0.333·9-s + 0.345·10-s + 1.05·11-s + 0.233·12-s − 0.914·13-s + 0.0619·14-s − 0.258·15-s − 0.432·16-s − 0.0903·17-s + 0.257·18-s − 0.846·19-s − 0.180·20-s − 0.0463·21-s + 0.811·22-s + 0.755·23-s + 0.625·24-s + 0.200·25-s − 0.705·26-s − 0.192·27-s − 0.0324·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 7 | \( 1 - 0.212T + 7T^{2} \) |
| 11 | \( 1 - 3.48T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 + 0.372T + 17T^{2} \) |
| 19 | \( 1 + 3.68T + 19T^{2} \) |
| 23 | \( 1 - 3.62T + 23T^{2} \) |
| 29 | \( 1 + 3.97T + 29T^{2} \) |
| 31 | \( 1 - 1.86T + 31T^{2} \) |
| 37 | \( 1 - 6.08T + 37T^{2} \) |
| 41 | \( 1 - 3.34T + 41T^{2} \) |
| 43 | \( 1 + 7.75T + 43T^{2} \) |
| 47 | \( 1 - 5.29T + 47T^{2} \) |
| 53 | \( 1 + 3.04T + 53T^{2} \) |
| 59 | \( 1 - 3.10T + 59T^{2} \) |
| 61 | \( 1 - 5.45T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 8.71T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 5.98T + 83T^{2} \) |
| 89 | \( 1 + 6.80T + 89T^{2} \) |
| 97 | \( 1 - 8.14T + 97T^{2} \) |
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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
−7.52870538905326599751970299692, −6.76444773726507766797704365072, −6.12899902707436460162961840693, −5.57601160550532951230665436748, −4.67854908035106818380900026855, −4.37095896051035160446576793010, −3.39659335549232043965889419272, −2.46116985788628894872553909309, −1.31132689362643101819049758504, 0,
1.31132689362643101819049758504, 2.46116985788628894872553909309, 3.39659335549232043965889419272, 4.37095896051035160446576793010, 4.67854908035106818380900026855, 5.57601160550532951230665436748, 6.12899902707436460162961840693, 6.76444773726507766797704365072, 7.52870538905326599751970299692
