L(s) = 1 | − 1.12·2-s + 0.383·3-s − 0.742·4-s + 5-s − 0.430·6-s + 1.34·7-s + 3.07·8-s − 2.85·9-s − 1.12·10-s + 1.85·11-s − 0.285·12-s − 0.128·13-s − 1.50·14-s + 0.383·15-s − 1.96·16-s + 17-s + 3.19·18-s + 1.93·19-s − 0.742·20-s + 0.516·21-s − 2.08·22-s + 0.872·23-s + 1.18·24-s + 25-s + 0.144·26-s − 2.24·27-s − 0.998·28-s + ⋯ |
L(s) = 1 | − 0.792·2-s + 0.221·3-s − 0.371·4-s + 0.447·5-s − 0.175·6-s + 0.508·7-s + 1.08·8-s − 0.950·9-s − 0.354·10-s + 0.560·11-s − 0.0822·12-s − 0.0356·13-s − 0.403·14-s + 0.0991·15-s − 0.490·16-s + 0.242·17-s + 0.754·18-s + 0.443·19-s − 0.166·20-s + 0.112·21-s − 0.444·22-s + 0.181·23-s + 0.240·24-s + 0.200·25-s + 0.0282·26-s − 0.432·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 71 | \( 1 + T \) |
good | 2 | \( 1 + 1.12T + 2T^{2} \) |
| 3 | \( 1 - 0.383T + 3T^{2} \) |
| 7 | \( 1 - 1.34T + 7T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 13 | \( 1 + 0.128T + 13T^{2} \) |
| 19 | \( 1 - 1.93T + 19T^{2} \) |
| 23 | \( 1 - 0.872T + 23T^{2} \) |
| 29 | \( 1 + 8.06T + 29T^{2} \) |
| 31 | \( 1 + 0.702T + 31T^{2} \) |
| 37 | \( 1 + 3.28T + 37T^{2} \) |
| 41 | \( 1 + 0.772T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 - 9.60T + 47T^{2} \) |
| 53 | \( 1 + 2.79T + 53T^{2} \) |
| 59 | \( 1 + 7.92T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 1.83T + 67T^{2} \) |
| 73 | \( 1 + 16.5T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 0.0156T + 89T^{2} \) |
| 97 | \( 1 - 18.4T + 97T^{2} \) |
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
−7.82607072743879342983484001359, −7.31705921172961404201628748928, −6.35348072425502012284484746075, −5.47624712133499469670517651017, −4.99333117067013987570544032668, −3.97902547331589964086035435599, −3.19693475516495074442141990315, −2.03647047254962616645114179932, −1.28761142360305209041041679560, 0,
1.28761142360305209041041679560, 2.03647047254962616645114179932, 3.19693475516495074442141990315, 3.97902547331589964086035435599, 4.99333117067013987570544032668, 5.47624712133499469670517651017, 6.35348072425502012284484746075, 7.31705921172961404201628748928, 7.82607072743879342983484001359