| L(s) = 1 | − 4.53·2-s − 9·3-s − 11.4·4-s − 46.1·5-s + 40.7·6-s − 25.4·7-s + 196.·8-s + 81·9-s + 209.·10-s − 248.·11-s + 103.·12-s − 109.·13-s + 115.·14-s + 415.·15-s − 526.·16-s − 679.·17-s − 367.·18-s − 1.97e3·19-s + 528.·20-s + 229.·21-s + 1.12e3·22-s − 2.50e3·23-s − 1.77e3·24-s − 992.·25-s + 496.·26-s − 729·27-s + 291.·28-s + ⋯ |
| L(s) = 1 | − 0.801·2-s − 0.577·3-s − 0.357·4-s − 0.826·5-s + 0.462·6-s − 0.196·7-s + 1.08·8-s + 0.333·9-s + 0.661·10-s − 0.618·11-s + 0.206·12-s − 0.179·13-s + 0.157·14-s + 0.476·15-s − 0.513·16-s − 0.570·17-s − 0.267·18-s − 1.25·19-s + 0.295·20-s + 0.113·21-s + 0.495·22-s − 0.988·23-s − 0.628·24-s − 0.317·25-s + 0.143·26-s − 0.192·27-s + 0.0702·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 309 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.06343665264\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06343665264\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 103 | \( 1 - 1.06e4T \) |
| good | 2 | \( 1 + 4.53T + 32T^{2} \) |
| 5 | \( 1 + 46.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 25.4T + 1.68e4T^{2} \) |
| 11 | \( 1 + 248.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 109.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 679.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.97e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.50e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.04e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.72e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 734.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.77e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 8.40e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.85e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.63e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.92e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.60e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.08e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.38e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.42e4T + 8.58e9T^{2} \) |
| 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
−10.78196496959900113047384879053, −9.944168012074872906879943956767, −8.959442530342478130157229596810, −7.995871314097169410423822648685, −7.30740767202545866915886459198, −5.98090317855148323686797846177, −4.69866461616187148337372916547, −3.83485487892394916682842383826, −1.90208313671515814343077800900, −0.15901501466223376559062001157,
0.15901501466223376559062001157, 1.90208313671515814343077800900, 3.83485487892394916682842383826, 4.69866461616187148337372916547, 5.98090317855148323686797846177, 7.30740767202545866915886459198, 7.995871314097169410423822648685, 8.959442530342478130157229596810, 9.944168012074872906879943956767, 10.78196496959900113047384879053
