| L(s) = 1 | + 3.23·3-s − 5-s + 7.47·9-s + 0.763·11-s − 13-s − 3.23·15-s + 2·17-s + 0.763·19-s − 3.23·23-s + 25-s + 14.4·27-s + 8.47·29-s − 5.70·31-s + 2.47·33-s − 8.47·37-s − 3.23·39-s − 10.9·41-s − 3.23·43-s − 7.47·45-s + 12.9·47-s − 7·49-s + 6.47·51-s − 10.9·53-s − 0.763·55-s + 2.47·57-s − 5.70·59-s − 4.47·61-s + ⋯ |
| L(s) = 1 | + 1.86·3-s − 0.447·5-s + 2.49·9-s + 0.230·11-s − 0.277·13-s − 0.835·15-s + 0.485·17-s + 0.175·19-s − 0.674·23-s + 0.200·25-s + 2.78·27-s + 1.57·29-s − 1.02·31-s + 0.430·33-s − 1.39·37-s − 0.518·39-s − 1.70·41-s − 0.493·43-s − 1.11·45-s + 1.88·47-s − 49-s + 0.906·51-s − 1.50·53-s − 0.103·55-s + 0.327·57-s − 0.743·59-s − 0.572·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.498900004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.498900004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 3.23T + 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 0.763T + 11T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 0.763T + 19T^{2} \) |
| 23 | \( 1 + 3.23T + 23T^{2} \) |
| 29 | \( 1 - 8.47T + 29T^{2} \) |
| 31 | \( 1 + 5.70T + 31T^{2} \) |
| 37 | \( 1 + 8.47T + 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 + 3.23T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 5.70T + 59T^{2} \) |
| 61 | \( 1 + 4.47T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 - 6.47T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.56278378945764439696970589015, −9.809808715310844001856216277415, −8.952511189705801405275600740161, −8.246040202668648446777859432296, −7.54526762183100646605203176810, −6.65197439433086180665809822993, −4.90408198265988296098013593211, −3.78543256608435766297839347422, −3.03037032440344790696608194379, −1.73464909987802140068195528826,
1.73464909987802140068195528826, 3.03037032440344790696608194379, 3.78543256608435766297839347422, 4.90408198265988296098013593211, 6.65197439433086180665809822993, 7.54526762183100646605203176810, 8.246040202668648446777859432296, 8.952511189705801405275600740161, 9.809808715310844001856216277415, 10.56278378945764439696970589015
