L(s) = 1 | + 2-s − 1.33·3-s + 4-s + 5-s − 1.33·6-s − 0.356·7-s + 8-s − 1.20·9-s + 10-s + 11-s − 1.33·12-s + 3.42·13-s − 0.356·14-s − 1.33·15-s + 16-s + 4.44·17-s − 1.20·18-s − 0.263·19-s + 20-s + 0.477·21-s + 22-s − 1.63·23-s − 1.33·24-s + 25-s + 3.42·26-s + 5.63·27-s − 0.356·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.773·3-s + 0.5·4-s + 0.447·5-s − 0.546·6-s − 0.134·7-s + 0.353·8-s − 0.402·9-s + 0.316·10-s + 0.301·11-s − 0.386·12-s + 0.950·13-s − 0.0952·14-s − 0.345·15-s + 0.250·16-s + 1.07·17-s − 0.284·18-s − 0.0605·19-s + 0.223·20-s + 0.104·21-s + 0.213·22-s − 0.341·23-s − 0.273·24-s + 0.200·25-s + 0.671·26-s + 1.08·27-s − 0.0673·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.613404378\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.613404378\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 1.33T + 3T^{2} \) |
| 7 | \( 1 + 0.356T + 7T^{2} \) |
| 13 | \( 1 - 3.42T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + 0.263T + 19T^{2} \) |
| 23 | \( 1 + 1.63T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 - 7.63T + 41T^{2} \) |
| 47 | \( 1 - 6.20T + 47T^{2} \) |
| 53 | \( 1 + 5.75T + 53T^{2} \) |
| 59 | \( 1 + 6.11T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 0.541T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 5.07T + 89T^{2} \) |
| 97 | \( 1 - 6.53T + 97T^{2} \) |
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
−8.235266775347256813902193946269, −7.40246391650763433397591560909, −6.43132275642885151380395544761, −6.10343694852543194695430987949, −5.43134986841003251702600495794, −4.78476164805652228176122723921, −3.73075181279707753785346123755, −3.09812330582270080311692621794, −1.93520104703070196093022463452, −0.857121845332315242430450258225,
0.857121845332315242430450258225, 1.93520104703070196093022463452, 3.09812330582270080311692621794, 3.73075181279707753785346123755, 4.78476164805652228176122723921, 5.43134986841003251702600495794, 6.10343694852543194695430987949, 6.43132275642885151380395544761, 7.40246391650763433397591560909, 8.235266775347256813902193946269