| L(s) = 1 | + 2-s + 4-s + 1.23·5-s − 7-s + 8-s + 1.23·10-s − 5.23·11-s − 3.23·13-s − 14-s + 16-s − 5.70·17-s + 19-s + 1.23·20-s − 5.23·22-s − 0.763·23-s − 3.47·25-s − 3.23·26-s − 28-s + 0.472·29-s − 2·31-s + 32-s − 5.70·34-s − 1.23·35-s − 1.52·37-s + 38-s + 1.23·40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.552·5-s − 0.377·7-s + 0.353·8-s + 0.390·10-s − 1.57·11-s − 0.897·13-s − 0.267·14-s + 0.250·16-s − 1.38·17-s + 0.229·19-s + 0.276·20-s − 1.11·22-s − 0.159·23-s − 0.694·25-s − 0.634·26-s − 0.188·28-s + 0.0876·29-s − 0.359·31-s + 0.176·32-s − 0.978·34-s − 0.208·35-s − 0.251·37-s + 0.162·38-s + 0.195·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 5.70T + 17T^{2} \) |
| 23 | \( 1 + 0.763T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 1.52T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 8.94T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 + 3.52T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 8.47T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 + 1.52T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 - 10T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 9.23T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.509490089403922826885863925464, −7.66521425348268400119782361046, −6.93486554432045261503408578900, −6.15670956721717250511281758921, −5.27743406657756424019839200940, −4.80943312274269754481010158797, −3.66389613637644212286254484191, −2.61169322932800022833582448284, −2.03572659756822336222215963540, 0,
2.03572659756822336222215963540, 2.61169322932800022833582448284, 3.66389613637644212286254484191, 4.80943312274269754481010158797, 5.27743406657756424019839200940, 6.15670956721717250511281758921, 6.93486554432045261503408578900, 7.66521425348268400119782361046, 8.509490089403922826885863925464
