| L(s) = 1 | + 2-s + 1.08·3-s + 4-s + 1.92·5-s + 1.08·6-s − 2.20·7-s + 8-s − 1.81·9-s + 1.92·10-s + 2.21·11-s + 1.08·12-s + 3.60·13-s − 2.20·14-s + 2.09·15-s + 16-s − 0.413·17-s − 1.81·18-s + 19-s + 1.92·20-s − 2.39·21-s + 2.21·22-s − 2.20·23-s + 1.08·24-s − 1.28·25-s + 3.60·26-s − 5.23·27-s − 2.20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.628·3-s + 0.5·4-s + 0.861·5-s + 0.444·6-s − 0.833·7-s + 0.353·8-s − 0.605·9-s + 0.609·10-s + 0.668·11-s + 0.314·12-s + 0.998·13-s − 0.589·14-s + 0.541·15-s + 0.250·16-s − 0.100·17-s − 0.428·18-s + 0.229·19-s + 0.430·20-s − 0.523·21-s + 0.472·22-s − 0.459·23-s + 0.222·24-s − 0.257·25-s + 0.706·26-s − 1.00·27-s − 0.416·28-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
| \(L(1)\) |
\(\approx\) |
\(4.764986693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.764986693\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 - 1.92T + 5T^{2} \) |
| 7 | \( 1 + 2.20T + 7T^{2} \) |
| 11 | \( 1 - 2.21T + 11T^{2} \) |
| 13 | \( 1 - 3.60T + 13T^{2} \) |
| 17 | \( 1 + 0.413T + 17T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 - 6.75T + 29T^{2} \) |
| 31 | \( 1 + 1.35T + 31T^{2} \) |
| 37 | \( 1 - 9.83T + 37T^{2} \) |
| 41 | \( 1 + 9.18T + 41T^{2} \) |
| 43 | \( 1 - 2.75T + 43T^{2} \) |
| 47 | \( 1 - 3.76T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 - 9.09T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 - 0.769T + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 - 2.35T + 79T^{2} \) |
| 83 | \( 1 - 8.42T + 83T^{2} \) |
| 89 | \( 1 + 3.48T + 89T^{2} \) |
| 97 | \( 1 - 2.48T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.897925819843446838300724718355, −6.82351572732913024954331007525, −6.37762081713096592065953507548, −5.82052496233963666133818276554, −5.17858580778172891832381283156, −3.97521555416165472653049593634, −3.60856660365808818091882349893, −2.69362604052975509188892994030, −2.12112066129329695276921959919, −0.956135316839507678812693663359,
0.956135316839507678812693663359, 2.12112066129329695276921959919, 2.69362604052975509188892994030, 3.60856660365808818091882349893, 3.97521555416165472653049593634, 5.17858580778172891832381283156, 5.82052496233963666133818276554, 6.37762081713096592065953507548, 6.82351572732913024954331007525, 7.897925819843446838300724718355
