| L(s) = 1 | + (−1.68 + 0.410i)3-s − 5-s + 4.25i·7-s + (2.66 − 1.38i)9-s − 1.77·11-s + 4.14·13-s + (1.68 − 0.410i)15-s + 6.86·17-s + 1.46i·19-s + (−1.75 − 7.16i)21-s + (−1.73 − 4.47i)23-s + 25-s + (−3.91 + 3.42i)27-s + 4.10i·29-s − 5.05·31-s + ⋯ |
| L(s) = 1 | + (−0.971 + 0.237i)3-s − 0.447·5-s + 1.60i·7-s + (0.887 − 0.461i)9-s − 0.533·11-s + 1.15·13-s + (0.434 − 0.106i)15-s + 1.66·17-s + 0.336i·19-s + (−0.381 − 1.56i)21-s + (−0.360 − 0.932i)23-s + 0.200·25-s + (−0.752 + 0.658i)27-s + 0.761i·29-s − 0.908·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8811544218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8811544218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.68 - 0.410i)T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + (1.73 + 4.47i)T \) |
| good | 7 | \( 1 - 4.25iT - 7T^{2} \) |
| 11 | \( 1 + 1.77T + 11T^{2} \) |
| 13 | \( 1 - 4.14T + 13T^{2} \) |
| 17 | \( 1 - 6.86T + 17T^{2} \) |
| 19 | \( 1 - 1.46iT - 19T^{2} \) |
| 29 | \( 1 - 4.10iT - 29T^{2} \) |
| 31 | \( 1 + 5.05T + 31T^{2} \) |
| 37 | \( 1 - 2.88iT - 37T^{2} \) |
| 41 | \( 1 - 4.56iT - 41T^{2} \) |
| 43 | \( 1 - 3.27iT - 43T^{2} \) |
| 47 | \( 1 + 5.80iT - 47T^{2} \) |
| 53 | \( 1 - 0.831T + 53T^{2} \) |
| 59 | \( 1 - 11.8iT - 59T^{2} \) |
| 61 | \( 1 - 12.0iT - 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 16.3iT - 71T^{2} \) |
| 73 | \( 1 + 1.69T + 73T^{2} \) |
| 79 | \( 1 - 2.62iT - 79T^{2} \) |
| 83 | \( 1 + 5.22T + 83T^{2} \) |
| 89 | \( 1 + 9.78T + 89T^{2} \) |
| 97 | \( 1 + 9.26iT - 97T^{2} \) |
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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
−10.01566364277742379748856778922, −8.993637207234595154168599634739, −8.348702886912131717067488239193, −7.43152168241778643103707837977, −6.27477489533842841042089202136, −5.69832224740870958605115449862, −5.08826834901274815937240601124, −3.89558201896117508319592149474, −2.88460728142602011396530277650, −1.34931469607878202591555992460,
0.47014177356967335437398415109, 1.49776833011542871026124011976, 3.48977118080083530150904029110, 4.07797610080567082461700296643, 5.19153703086495563657934873044, 5.95576258016168910601903854472, 6.96986089663543191016635681858, 7.59274883042848226039614639247, 8.124660054029537827058802898845, 9.601809125160468021874107196404
