| L(s) = 1 | − 2-s + 1.61·3-s + 4-s − 2.61·5-s − 1.61·6-s + 7-s − 8-s − 0.381·9-s + 2.61·10-s − 4.85·11-s + 1.61·12-s − 4.47·13-s − 14-s − 4.23·15-s + 16-s + 0.763·17-s + 0.381·18-s − 2.61·20-s + 1.61·21-s + 4.85·22-s + 8.94·23-s − 1.61·24-s + 1.85·25-s + 4.47·26-s − 5.47·27-s + 28-s + 0.145·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.934·3-s + 0.5·4-s − 1.17·5-s − 0.660·6-s + 0.377·7-s − 0.353·8-s − 0.127·9-s + 0.827·10-s − 1.46·11-s + 0.467·12-s − 1.24·13-s − 0.267·14-s − 1.09·15-s + 0.250·16-s + 0.185·17-s + 0.0900·18-s − 0.585·20-s + 0.353·21-s + 1.03·22-s + 1.86·23-s − 0.330·24-s + 0.370·25-s + 0.877·26-s − 1.05·27-s + 0.188·28-s + 0.0270·29-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
| \(L(1)\) |
\(\approx\) |
\(0.9348070172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9348070172\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 2.61T + 5T^{2} \) |
| 11 | \( 1 + 4.85T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 0.763T + 17T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 - 0.145T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 7.85T + 37T^{2} \) |
| 41 | \( 1 + 9.56T + 41T^{2} \) |
| 43 | \( 1 - 3.85T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 5.14T + 53T^{2} \) |
| 59 | \( 1 + 11.5T + 59T^{2} \) |
| 61 | \( 1 + 8.56T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 - 8.56T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 0.326T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 3.14T + 89T^{2} \) |
| 97 | \( 1 - 7.14T + 97T^{2} \) |
| 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
−8.188036296471928122640384861596, −7.61141231546193969286285753974, −7.40085483958602519693866553164, −6.27883078710622879192025017049, −5.04646806608112318358805159491, −4.66859573531801999687659414156, −3.27135575430449795238275921617, −2.93208619565584855613044444588, −2.04147363234434574697612834699, −0.52957002926456084968500642065,
0.52957002926456084968500642065, 2.04147363234434574697612834699, 2.93208619565584855613044444588, 3.27135575430449795238275921617, 4.66859573531801999687659414156, 5.04646806608112318358805159491, 6.27883078710622879192025017049, 7.40085483958602519693866553164, 7.61141231546193969286285753974, 8.188036296471928122640384861596
