L(s) = 1 | − 2-s − 0.918·3-s + 4-s − 5-s + 0.918·6-s − 3.92·7-s − 8-s − 2.15·9-s + 10-s + 1.54·11-s − 0.918·12-s − 0.562·13-s + 3.92·14-s + 0.918·15-s + 16-s − 6.59·17-s + 2.15·18-s + 6.11·19-s − 20-s + 3.60·21-s − 1.54·22-s + 4.55·23-s + 0.918·24-s + 25-s + 0.562·26-s + 4.73·27-s − 3.92·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.530·3-s + 0.5·4-s − 0.447·5-s + 0.375·6-s − 1.48·7-s − 0.353·8-s − 0.718·9-s + 0.316·10-s + 0.465·11-s − 0.265·12-s − 0.155·13-s + 1.04·14-s + 0.237·15-s + 0.250·16-s − 1.60·17-s + 0.508·18-s + 1.40·19-s − 0.223·20-s + 0.787·21-s − 0.329·22-s + 0.949·23-s + 0.187·24-s + 0.200·25-s + 0.110·26-s + 0.911·27-s − 0.741·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 0.918T + 3T^{2} \) |
| 7 | \( 1 + 3.92T + 7T^{2} \) |
| 11 | \( 1 - 1.54T + 11T^{2} \) |
| 13 | \( 1 + 0.562T + 13T^{2} \) |
| 17 | \( 1 + 6.59T + 17T^{2} \) |
| 19 | \( 1 - 6.11T + 19T^{2} \) |
| 23 | \( 1 - 4.55T + 23T^{2} \) |
| 29 | \( 1 - 4.69T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 + 2.86T + 37T^{2} \) |
| 41 | \( 1 - 6.29T + 41T^{2} \) |
| 43 | \( 1 + 3.87T + 43T^{2} \) |
| 47 | \( 1 + 2.99T + 47T^{2} \) |
| 53 | \( 1 - 8.81T + 53T^{2} \) |
| 59 | \( 1 - 8.63T + 59T^{2} \) |
| 61 | \( 1 + 5.01T + 61T^{2} \) |
| 67 | \( 1 - 1.74T + 67T^{2} \) |
| 71 | \( 1 + 7.67T + 71T^{2} \) |
| 73 | \( 1 - 3.93T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 4.99T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 4.83T + 97T^{2} \) |
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
−8.258205930076386872130956675374, −7.08044233719543860585794011897, −6.78910715491639302825140078340, −6.08969005095913684494101883601, −5.24137352388360780168128785949, −4.22123687030090551656072482364, −3.14567038879806148831207899276, −2.64241627226355832816110911206, −0.971018027044129258548824435129, 0,
0.971018027044129258548824435129, 2.64241627226355832816110911206, 3.14567038879806148831207899276, 4.22123687030090551656072482364, 5.24137352388360780168128785949, 6.08969005095913684494101883601, 6.78910715491639302825140078340, 7.08044233719543860585794011897, 8.258205930076386872130956675374