| L(s) = 1 | − 2-s + 3.17·3-s + 4-s − 1.33·5-s − 3.17·6-s + 1.08·7-s − 8-s + 7.10·9-s + 1.33·10-s + 3.46·11-s + 3.17·12-s + 2.96·13-s − 1.08·14-s − 4.24·15-s + 16-s + 5.49·17-s − 7.10·18-s − 19-s − 1.33·20-s + 3.45·21-s − 3.46·22-s + 8.13·23-s − 3.17·24-s − 3.21·25-s − 2.96·26-s + 13.0·27-s + 1.08·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.83·3-s + 0.5·4-s − 0.596·5-s − 1.29·6-s + 0.410·7-s − 0.353·8-s + 2.36·9-s + 0.421·10-s + 1.04·11-s + 0.917·12-s + 0.823·13-s − 0.290·14-s − 1.09·15-s + 0.250·16-s + 1.33·17-s − 1.67·18-s − 0.229·19-s − 0.298·20-s + 0.753·21-s − 0.738·22-s + 1.69·23-s − 0.648·24-s − 0.643·25-s − 0.582·26-s + 2.50·27-s + 0.205·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.793250316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.793250316\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 3.17T + 3T^{2} \) |
| 5 | \( 1 + 1.33T + 5T^{2} \) |
| 7 | \( 1 - 1.08T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 2.96T + 13T^{2} \) |
| 17 | \( 1 - 5.49T + 17T^{2} \) |
| 23 | \( 1 - 8.13T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 - 8.74T + 31T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 1.54T + 43T^{2} \) |
| 47 | \( 1 + 6.03T + 47T^{2} \) |
| 53 | \( 1 + 1.06T + 53T^{2} \) |
| 59 | \( 1 + 12.8T + 59T^{2} \) |
| 61 | \( 1 + 8.17T + 61T^{2} \) |
| 67 | \( 1 - 0.632T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 1.72T + 79T^{2} \) |
| 83 | \( 1 + 3.81T + 83T^{2} \) |
| 89 | \( 1 + 3.11T + 89T^{2} \) |
| 97 | \( 1 - 2.24T + 97T^{2} \) |
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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
−7.899143880941963056134042212293, −7.56302166852471519526843792905, −6.77579626004219183399724843000, −6.03910161066152423445690549939, −4.67849864357430319110903113464, −4.08050526392637780942634771780, −3.18786147693187798191286094816, −2.89855670886560330328789993176, −1.54657460040693248277592997969, −1.16352838231869752859260981095,
1.16352838231869752859260981095, 1.54657460040693248277592997969, 2.89855670886560330328789993176, 3.18786147693187798191286094816, 4.08050526392637780942634771780, 4.67849864357430319110903113464, 6.03910161066152423445690549939, 6.77579626004219183399724843000, 7.56302166852471519526843792905, 7.899143880941963056134042212293
