| L(s) = 1 | − 2-s + 4-s + 3.73·5-s − 7-s − 8-s − 3.73·10-s − 4.19·11-s + 0.464·13-s + 14-s + 16-s + 7·17-s − 2.73·19-s + 3.73·20-s + 4.19·22-s + 6.19·23-s + 8.92·25-s − 0.464·26-s − 28-s + 8.46·29-s − 2.19·31-s − 32-s − 7·34-s − 3.73·35-s − 6.66·37-s + 2.73·38-s − 3.73·40-s + 9.46·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.66·5-s − 0.377·7-s − 0.353·8-s − 1.18·10-s − 1.26·11-s + 0.128·13-s + 0.267·14-s + 0.250·16-s + 1.69·17-s − 0.626·19-s + 0.834·20-s + 0.894·22-s + 1.29·23-s + 1.78·25-s − 0.0910·26-s − 0.188·28-s + 1.57·29-s − 0.394·31-s − 0.176·32-s − 1.20·34-s − 0.630·35-s − 1.09·37-s + 0.443·38-s − 0.590·40-s + 1.47·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.527058950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.527058950\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 - 3.73T + 5T^{2} \) |
| 11 | \( 1 + 4.19T + 11T^{2} \) |
| 13 | \( 1 - 0.464T + 13T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 23 | \( 1 - 6.19T + 23T^{2} \) |
| 29 | \( 1 - 8.46T + 29T^{2} \) |
| 31 | \( 1 + 2.19T + 31T^{2} \) |
| 37 | \( 1 + 6.66T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 - 5.46T + 43T^{2} \) |
| 47 | \( 1 - 1.26T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 + 6.19T + 59T^{2} \) |
| 61 | \( 1 + 9.92T + 61T^{2} \) |
| 67 | \( 1 + 3.26T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 + 3.92T + 89T^{2} \) |
| 97 | \( 1 + 2.92T + 97T^{2} \) |
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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
−9.764491109229933810634573918241, −9.214678306585608311542763007920, −8.289468455539870886670604642057, −7.39123445470273608523883788881, −6.42458775102284963770257394605, −5.70335391592220612498138019709, −4.98664153712767081207825785621, −3.13081215238899425447998646359, −2.37442263818242509803958140776, −1.09808247313066073318935895647,
1.09808247313066073318935895647, 2.37442263818242509803958140776, 3.13081215238899425447998646359, 4.98664153712767081207825785621, 5.70335391592220612498138019709, 6.42458775102284963770257394605, 7.39123445470273608523883788881, 8.289468455539870886670604642057, 9.214678306585608311542763007920, 9.764491109229933810634573918241
