| L(s) = 1 | + 2.74·2-s − 3-s + 5.52·4-s − 2.74·6-s − 1.02·7-s + 9.68·8-s + 9-s − 0.119·11-s − 5.52·12-s + 1.18·13-s − 2.82·14-s + 15.5·16-s + 3.82·17-s + 2.74·18-s + 0.0209·19-s + 1.02·21-s − 0.327·22-s + 1.16·23-s − 9.68·24-s + 3.24·26-s − 27-s − 5.69·28-s − 7.97·29-s + 8.11·31-s + 23.1·32-s + 0.119·33-s + 10.4·34-s + ⋯ |
| L(s) = 1 | + 1.94·2-s − 0.577·3-s + 2.76·4-s − 1.12·6-s − 0.389·7-s + 3.42·8-s + 0.333·9-s − 0.0360·11-s − 1.59·12-s + 0.328·13-s − 0.754·14-s + 3.87·16-s + 0.927·17-s + 0.646·18-s + 0.00480·19-s + 0.224·21-s − 0.0698·22-s + 0.243·23-s − 1.97·24-s + 0.636·26-s − 0.192·27-s − 1.07·28-s − 1.48·29-s + 1.45·31-s + 4.09·32-s + 0.0207·33-s + 1.79·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.023798097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.023798097\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 7 | \( 1 + 1.02T + 7T^{2} \) |
| 11 | \( 1 + 0.119T + 11T^{2} \) |
| 13 | \( 1 - 1.18T + 13T^{2} \) |
| 17 | \( 1 - 3.82T + 17T^{2} \) |
| 19 | \( 1 - 0.0209T + 19T^{2} \) |
| 23 | \( 1 - 1.16T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 - 8.11T + 31T^{2} \) |
| 37 | \( 1 - 5.42T + 37T^{2} \) |
| 41 | \( 1 - 1.83T + 41T^{2} \) |
| 43 | \( 1 - 1.46T + 43T^{2} \) |
| 53 | \( 1 + 4.92T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 2.94T + 61T^{2} \) |
| 67 | \( 1 - 5.46T + 67T^{2} \) |
| 71 | \( 1 + 7.27T + 71T^{2} \) |
| 73 | \( 1 - 13.6T + 73T^{2} \) |
| 79 | \( 1 + 9.77T + 79T^{2} \) |
| 83 | \( 1 + 2.67T + 83T^{2} \) |
| 89 | \( 1 - 1.54T + 89T^{2} \) |
| 97 | \( 1 + 0.676T + 97T^{2} \) |
| 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
−8.197309674378310158321016094562, −7.45007267825880476277683512599, −6.72244785459226238924333131297, −6.07572388212887068798902521755, −5.52997567386651697596109707824, −4.80408424512041241685870673461, −3.99282263586458520926766584178, −3.30953378889317122394385057218, −2.41034132936887952999522160601, −1.21886947335270998255421917943,
1.21886947335270998255421917943, 2.41034132936887952999522160601, 3.30953378889317122394385057218, 3.99282263586458520926766584178, 4.80408424512041241685870673461, 5.52997567386651697596109707824, 6.07572388212887068798902521755, 6.72244785459226238924333131297, 7.45007267825880476277683512599, 8.197309674378310158321016094562
