| L(s) = 1 | + 2-s + 2.93·3-s + 4-s − 0.167·5-s + 2.93·6-s + 7-s + 8-s + 5.62·9-s − 0.167·10-s + 5.85·11-s + 2.93·12-s + 1.66·13-s + 14-s − 0.493·15-s + 16-s − 1.41·17-s + 5.62·18-s − 0.167·20-s + 2.93·21-s + 5.85·22-s + 8.26·23-s + 2.93·24-s − 4.97·25-s + 1.66·26-s + 7.71·27-s + 28-s − 6.89·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.69·3-s + 0.5·4-s − 0.0750·5-s + 1.19·6-s + 0.377·7-s + 0.353·8-s + 1.87·9-s − 0.0530·10-s + 1.76·11-s + 0.847·12-s + 0.460·13-s + 0.267·14-s − 0.127·15-s + 0.250·16-s − 0.343·17-s + 1.32·18-s − 0.0375·20-s + 0.640·21-s + 1.24·22-s + 1.72·23-s + 0.599·24-s − 0.994·25-s + 0.325·26-s + 1.48·27-s + 0.188·28-s − 1.28·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.949881448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.949881448\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.93T + 3T^{2} \) |
| 5 | \( 1 + 0.167T + 5T^{2} \) |
| 11 | \( 1 - 5.85T + 11T^{2} \) |
| 13 | \( 1 - 1.66T + 13T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 23 | \( 1 - 8.26T + 23T^{2} \) |
| 29 | \( 1 + 6.89T + 29T^{2} \) |
| 31 | \( 1 + 8.06T + 31T^{2} \) |
| 37 | \( 1 + 6.18T + 37T^{2} \) |
| 41 | \( 1 + 4.84T + 41T^{2} \) |
| 43 | \( 1 - 1.16T + 43T^{2} \) |
| 47 | \( 1 - 0.154T + 47T^{2} \) |
| 53 | \( 1 + 1.53T + 53T^{2} \) |
| 59 | \( 1 + 5.20T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 2.21T + 67T^{2} \) |
| 71 | \( 1 + 8.24T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 5.69T + 79T^{2} \) |
| 83 | \( 1 + 9.93T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.337680400902800931599256119423, −7.30650915956305256333586597809, −7.10633438769988415283426306506, −6.12079301557280506738305625086, −5.14587743065430684648723853400, −4.14419175395476675030411775482, −3.71935251927612100312805536347, −3.11143128087627142433337617720, −1.95168925061246645292402807511, −1.45543287102080517003389958797,
1.45543287102080517003389958797, 1.95168925061246645292402807511, 3.11143128087627142433337617720, 3.71935251927612100312805536347, 4.14419175395476675030411775482, 5.14587743065430684648723853400, 6.12079301557280506738305625086, 7.10633438769988415283426306506, 7.30650915956305256333586597809, 8.337680400902800931599256119423
