| L(s) = 1 | + 2.82·3-s − 5-s + 2.09·7-s + 4.99·9-s − 4.99·13-s − 2.82·15-s + 2.80·19-s + 5.92·21-s − 8.45·23-s + 25-s + 5.62·27-s − 4·29-s − 9.18·31-s − 2.09·35-s − 9.98·37-s − 14.1·39-s + 2.53·41-s + 0.233·43-s − 4.99·45-s − 9.15·47-s − 2.61·49-s + 1.33·53-s + 7.91·57-s − 12.1·59-s + 1.12·61-s + 10.4·63-s + 4.99·65-s + ⋯ |
| L(s) = 1 | + 1.63·3-s − 0.447·5-s + 0.791·7-s + 1.66·9-s − 1.38·13-s − 0.729·15-s + 0.642·19-s + 1.29·21-s − 1.76·23-s + 0.200·25-s + 1.08·27-s − 0.742·29-s − 1.64·31-s − 0.354·35-s − 1.64·37-s − 2.25·39-s + 0.395·41-s + 0.0356·43-s − 0.744·45-s − 1.33·47-s − 0.373·49-s + 0.183·53-s + 1.04·57-s − 1.57·59-s + 0.143·61-s + 1.31·63-s + 0.619·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 - 2.09T + 7T^{2} \) |
| 13 | \( 1 + 4.99T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 2.80T + 19T^{2} \) |
| 23 | \( 1 + 8.45T + 23T^{2} \) |
| 29 | \( 1 + 4T + 29T^{2} \) |
| 31 | \( 1 + 9.18T + 31T^{2} \) |
| 37 | \( 1 + 9.98T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 - 0.233T + 43T^{2} \) |
| 47 | \( 1 + 9.15T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 1.12T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 - 7.72T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 8.60T + 89T^{2} \) |
| 97 | \( 1 + 3.04T + 97T^{2} \) |
| 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
−7.39978757008716278154453316097, −7.23754813431319223230198700393, −5.94676033882238045205428003437, −5.05480355771892218049849775572, −4.44102239991572346583896289794, −3.62896965604636943885027106008, −3.12530124079580747050494071797, −2.03650857308754127556879998884, −1.76363794217171689242095748292, 0,
1.76363794217171689242095748292, 2.03650857308754127556879998884, 3.12530124079580747050494071797, 3.62896965604636943885027106008, 4.44102239991572346583896289794, 5.05480355771892218049849775572, 5.94676033882238045205428003437, 7.23754813431319223230198700393, 7.39978757008716278154453316097
