| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 0.618·7-s − 8-s − 2·9-s − 3·11-s + 12-s − 3.47·13-s − 0.618·14-s + 16-s + 7.85·17-s + 2·18-s − 2.76·19-s + 0.618·21-s + 3·22-s − 4.85·23-s − 24-s + 3.47·26-s − 5·27-s + 0.618·28-s − 6.70·29-s − 10.2·31-s − 32-s − 3·33-s − 7.85·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.233·7-s − 0.353·8-s − 0.666·9-s − 0.904·11-s + 0.288·12-s − 0.962·13-s − 0.165·14-s + 0.250·16-s + 1.90·17-s + 0.471·18-s − 0.634·19-s + 0.134·21-s + 0.639·22-s − 1.01·23-s − 0.204·24-s + 0.680·26-s − 0.962·27-s + 0.116·28-s − 1.24·29-s − 1.83·31-s − 0.176·32-s − 0.522·33-s − 1.34·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 - 0.618T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 - 7.85T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 + 4.85T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 - 3.70T + 41T^{2} \) |
| 43 | \( 1 + 0.909T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 + 0.708T + 53T^{2} \) |
| 59 | \( 1 - 6.70T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 1.23T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 - 1.85T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
| 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
−9.462874183603858738721839108054, −8.270948187845805088474156118251, −7.893618773295784762001528711615, −7.22735103750466004223517372787, −5.83929803336506889610783644325, −5.28843296477008836701795682304, −3.78374750057632773036392239499, −2.79371059366599156569172034683, −1.87518257534085333243675595322, 0,
1.87518257534085333243675595322, 2.79371059366599156569172034683, 3.78374750057632773036392239499, 5.28843296477008836701795682304, 5.83929803336506889610783644325, 7.22735103750466004223517372787, 7.893618773295784762001528711615, 8.270948187845805088474156118251, 9.462874183603858738721839108054
