| L(s) = 1 | − 2.67·2-s + 3-s + 5.15·4-s + 5-s − 2.67·6-s + 1.28·7-s − 8.44·8-s + 9-s − 2.67·10-s − 2.54·11-s + 5.15·12-s − 13-s − 3.44·14-s + 15-s + 12.2·16-s − 4.64·17-s − 2.67·18-s + 6.87·19-s + 5.15·20-s + 1.28·21-s + 6.81·22-s − 4.10·23-s − 8.44·24-s + 25-s + 2.67·26-s + 27-s + 6.63·28-s + ⋯ |
| L(s) = 1 | − 1.89·2-s + 0.577·3-s + 2.57·4-s + 0.447·5-s − 1.09·6-s + 0.486·7-s − 2.98·8-s + 0.333·9-s − 0.845·10-s − 0.768·11-s + 1.48·12-s − 0.277·13-s − 0.919·14-s + 0.258·15-s + 3.06·16-s − 1.12·17-s − 0.630·18-s + 1.57·19-s + 1.15·20-s + 0.280·21-s + 1.45·22-s − 0.856·23-s − 1.72·24-s + 0.200·25-s + 0.524·26-s + 0.192·27-s + 1.25·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 2.67T + 2T^{2} \) |
| 7 | \( 1 - 1.28T + 7T^{2} \) |
| 11 | \( 1 + 2.54T + 11T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 - 6.87T + 19T^{2} \) |
| 23 | \( 1 + 4.10T + 23T^{2} \) |
| 29 | \( 1 + 2.94T + 29T^{2} \) |
| 37 | \( 1 + 7.94T + 37T^{2} \) |
| 41 | \( 1 - 1.41T + 41T^{2} \) |
| 43 | \( 1 + 4.87T + 43T^{2} \) |
| 47 | \( 1 - 0.154T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 0.0417T + 59T^{2} \) |
| 61 | \( 1 - 9.68T + 61T^{2} \) |
| 67 | \( 1 - 9.78T + 67T^{2} \) |
| 71 | \( 1 - 1.01T + 71T^{2} \) |
| 73 | \( 1 + 4.64T + 73T^{2} \) |
| 79 | \( 1 + 0.226T + 79T^{2} \) |
| 83 | \( 1 - 1.23T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 7.38T + 97T^{2} \) |
| 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
−8.068871685333920989948259338266, −7.15934637521288582588191622722, −6.86781889918886771917810074849, −5.75058343999247982857032675570, −5.07017751134232993710641928651, −3.70098944805743148414776375164, −2.67438532152874386559086125969, −2.10643142738580510394404111290, −1.30613194375693015971823149844, 0,
1.30613194375693015971823149844, 2.10643142738580510394404111290, 2.67438532152874386559086125969, 3.70098944805743148414776375164, 5.07017751134232993710641928651, 5.75058343999247982857032675570, 6.86781889918886771917810074849, 7.15934637521288582588191622722, 8.068871685333920989948259338266
