| L(s) = 1 | − 1.73·2-s + 2·3-s + 0.999·4-s + 5-s − 3.46·6-s − 3.46·7-s + 1.73·8-s + 9-s − 1.73·10-s + 1.99·12-s + 5.99·14-s + 2·15-s − 5·16-s + 6.92·17-s − 1.73·18-s + 6.92·19-s + 0.999·20-s − 6.92·21-s + 6·23-s + 3.46·24-s + 25-s − 4·27-s − 3.46·28-s − 3.46·30-s + 4·31-s + 5.19·32-s − 11.9·34-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 1.15·3-s + 0.499·4-s + 0.447·5-s − 1.41·6-s − 1.30·7-s + 0.612·8-s + 0.333·9-s − 0.547·10-s + 0.577·12-s + 1.60·14-s + 0.516·15-s − 1.25·16-s + 1.68·17-s − 0.408·18-s + 1.58·19-s + 0.223·20-s − 1.51·21-s + 1.25·23-s + 0.707·24-s + 0.200·25-s − 0.769·27-s − 0.654·28-s − 0.632·30-s + 0.718·31-s + 0.918·32-s − 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.106009016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.106009016\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 - 2T + 3T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 - 3.46T + 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
| 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
−10.00117308451973018425458305294, −9.619189228809729584481617239192, −9.178728826564999716352724656186, −8.107558035291569018298785318844, −7.53268986741709245281207857225, −6.48342043254843083649398930624, −5.20416491656908695149759919949, −3.47888520076999818633201058568, −2.76248133346191418758358974995, −1.11646022545911958060614146128,
1.11646022545911958060614146128, 2.76248133346191418758358974995, 3.47888520076999818633201058568, 5.20416491656908695149759919949, 6.48342043254843083649398930624, 7.53268986741709245281207857225, 8.107558035291569018298785318844, 9.178728826564999716352724656186, 9.619189228809729584481617239192, 10.00117308451973018425458305294
