| L(s) = 1 | − 2-s − 2.73·3-s + 4-s + 5-s + 2.73·6-s − 1.26·7-s − 8-s + 4.46·9-s − 10-s + 1.46·11-s − 2.73·12-s + 1.46·13-s + 1.26·14-s − 2.73·15-s + 16-s − 1.46·17-s − 4.46·18-s − 4.19·19-s + 20-s + 3.46·21-s − 1.46·22-s − 8·23-s + 2.73·24-s + 25-s − 1.46·26-s − 3.99·27-s − 1.26·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.57·3-s + 0.5·4-s + 0.447·5-s + 1.11·6-s − 0.479·7-s − 0.353·8-s + 1.48·9-s − 0.316·10-s + 0.441·11-s − 0.788·12-s + 0.406·13-s + 0.338·14-s − 0.705·15-s + 0.250·16-s − 0.355·17-s − 1.05·18-s − 0.962·19-s + 0.223·20-s + 0.755·21-s − 0.312·22-s − 1.66·23-s + 0.557·24-s + 0.200·25-s − 0.287·26-s − 0.769·27-s − 0.239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 - 1.46T + 13T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 + 4.19T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 8.92T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 6.92T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 0.196T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 5.26T + 79T^{2} \) |
| 83 | \( 1 + 5.26T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
| 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
−10.96571294432696385833204859678, −10.10950998108117998920238357827, −9.380578784321491564422729633152, −8.157429517619222708056633769758, −6.77938727050772620060864557584, −6.26288012231895916120009879567, −5.38554572621642410432621906057, −3.92751481055437510594866452905, −1.81490457807024976965961734852, 0,
1.81490457807024976965961734852, 3.92751481055437510594866452905, 5.38554572621642410432621906057, 6.26288012231895916120009879567, 6.77938727050772620060864557584, 8.157429517619222708056633769758, 9.380578784321491564422729633152, 10.10950998108117998920238357827, 10.96571294432696385833204859678
