L(s) = 1 | − 0.833·3-s − 5-s − 1.87·7-s − 2.30·9-s − 0.792·11-s + 5.31·13-s + 0.833·15-s + 3.48·19-s + 1.56·21-s − 3.23·23-s + 25-s + 4.42·27-s − 1.45·29-s − 4.65·31-s + 0.660·33-s + 1.87·35-s − 1.52·37-s − 4.42·39-s + 5.84·41-s + 0.403·43-s + 2.30·45-s + 3.94·47-s − 3.47·49-s − 11.7·53-s + 0.792·55-s − 2.90·57-s + 10.9·59-s + ⋯ |
L(s) = 1 | − 0.481·3-s − 0.447·5-s − 0.709·7-s − 0.768·9-s − 0.239·11-s + 1.47·13-s + 0.215·15-s + 0.799·19-s + 0.341·21-s − 0.673·23-s + 0.200·25-s + 0.850·27-s − 0.270·29-s − 0.836·31-s + 0.115·33-s + 0.317·35-s − 0.250·37-s − 0.708·39-s + 0.912·41-s + 0.0615·43-s + 0.343·45-s + 0.575·47-s − 0.496·49-s − 1.61·53-s + 0.106·55-s − 0.384·57-s + 1.41·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 3 | \( 1 + 0.833T + 3T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 + 0.792T + 11T^{2} \) |
| 13 | \( 1 - 5.31T + 13T^{2} \) |
| 19 | \( 1 - 3.48T + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 + 1.45T + 29T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 + 1.52T + 37T^{2} \) |
| 41 | \( 1 - 5.84T + 41T^{2} \) |
| 43 | \( 1 - 0.403T + 43T^{2} \) |
| 47 | \( 1 - 3.94T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 + 6.08T + 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 - 6.48T + 73T^{2} \) |
| 79 | \( 1 - 7.26T + 79T^{2} \) |
| 83 | \( 1 + 4.94T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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
−7.906294264085752635816098493796, −6.90045555948094898650038483136, −6.28326277437241106189414451290, −5.68011475012298214682987612320, −5.02433419213215860627380200898, −3.83030091987415709798062176091, −3.44533714078215252123551294882, −2.44082649016923875836468697121, −1.09702075634939030350092262024, 0,
1.09702075634939030350092262024, 2.44082649016923875836468697121, 3.44533714078215252123551294882, 3.83030091987415709798062176091, 5.02433419213215860627380200898, 5.68011475012298214682987612320, 6.28326277437241106189414451290, 6.90045555948094898650038483136, 7.906294264085752635816098493796