| L(s) = 1 | + 2.25·2-s + 0.777·3-s + 3.09·4-s + 5-s + 1.75·6-s + 0.123·7-s + 2.48·8-s − 2.39·9-s + 2.25·10-s + 2.40·12-s + 5.49·13-s + 0.279·14-s + 0.777·15-s − 0.596·16-s + 0.519·17-s − 5.40·18-s + 3.15·19-s + 3.09·20-s + 0.0963·21-s − 7.92·23-s + 1.92·24-s + 25-s + 12.4·26-s − 4.19·27-s + 0.384·28-s − 4.07·29-s + 1.75·30-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.448·3-s + 1.54·4-s + 0.447·5-s + 0.716·6-s + 0.0468·7-s + 0.876·8-s − 0.798·9-s + 0.714·10-s + 0.695·12-s + 1.52·13-s + 0.0748·14-s + 0.200·15-s − 0.149·16-s + 0.125·17-s − 1.27·18-s + 0.724·19-s + 0.692·20-s + 0.0210·21-s − 1.65·23-s + 0.393·24-s + 0.200·25-s + 2.43·26-s − 0.807·27-s + 0.0725·28-s − 0.757·29-s + 0.320·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.058239132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.058239132\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.25T + 2T^{2} \) |
| 3 | \( 1 - 0.777T + 3T^{2} \) |
| 7 | \( 1 - 0.123T + 7T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 - 0.519T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 + 7.92T + 23T^{2} \) |
| 29 | \( 1 + 4.07T + 29T^{2} \) |
| 31 | \( 1 - 7.04T + 31T^{2} \) |
| 37 | \( 1 + 8.78T + 37T^{2} \) |
| 41 | \( 1 + 7.55T + 41T^{2} \) |
| 43 | \( 1 - 3.42T + 43T^{2} \) |
| 47 | \( 1 - 0.456T + 47T^{2} \) |
| 53 | \( 1 - 0.0354T + 53T^{2} \) |
| 59 | \( 1 + 5.47T + 59T^{2} \) |
| 61 | \( 1 - 7.68T + 61T^{2} \) |
| 67 | \( 1 + 2.53T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 8.52T + 73T^{2} \) |
| 79 | \( 1 - 6.27T + 79T^{2} \) |
| 83 | \( 1 - 0.626T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 + 1.83T + 97T^{2} \) |
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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
−11.01176399124925074686367121653, −9.885009571119870235646237543040, −8.789752981142532486862986954232, −7.985005519323449451212357437984, −6.59981833869442945804463241546, −5.91438795048445607648537405342, −5.17594273225296416952173819251, −3.86494629608583165998401050357, −3.19930343251278315123318221028, −1.95688892379934227145845556046,
1.95688892379934227145845556046, 3.19930343251278315123318221028, 3.86494629608583165998401050357, 5.17594273225296416952173819251, 5.91438795048445607648537405342, 6.59981833869442945804463241546, 7.985005519323449451212357437984, 8.789752981142532486862986954232, 9.885009571119870235646237543040, 11.01176399124925074686367121653
