| L(s) = 1 | − 0.305·2-s − 1.90·4-s + 1.87·7-s + 1.19·8-s + 5.53·11-s + 3.38·13-s − 0.571·14-s + 3.44·16-s + 6.94·17-s − 3.10·19-s − 1.69·22-s + 4.02·23-s − 1.03·26-s − 3.56·28-s + 4.73·29-s + 3.79·31-s − 3.44·32-s − 2.12·34-s + 7.24·37-s + 0.949·38-s + 4.81·41-s + 8.46·43-s − 10.5·44-s − 1.22·46-s − 6.19·47-s − 3.49·49-s − 6.45·52-s + ⋯ |
| L(s) = 1 | − 0.216·2-s − 0.953·4-s + 0.707·7-s + 0.422·8-s + 1.67·11-s + 0.939·13-s − 0.152·14-s + 0.862·16-s + 1.68·17-s − 0.712·19-s − 0.360·22-s + 0.838·23-s − 0.202·26-s − 0.674·28-s + 0.879·29-s + 0.681·31-s − 0.608·32-s − 0.363·34-s + 1.19·37-s + 0.153·38-s + 0.751·41-s + 1.29·43-s − 1.59·44-s − 0.181·46-s − 0.903·47-s − 0.499·49-s − 0.895·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.170507595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.170507595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.305T + 2T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 11 | \( 1 - 5.53T + 11T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 - 6.94T + 17T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 23 | \( 1 - 4.02T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 - 3.79T + 31T^{2} \) |
| 37 | \( 1 - 7.24T + 37T^{2} \) |
| 41 | \( 1 - 4.81T + 41T^{2} \) |
| 43 | \( 1 - 8.46T + 43T^{2} \) |
| 47 | \( 1 + 6.19T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 - 3.56T + 67T^{2} \) |
| 71 | \( 1 + 8.39T + 71T^{2} \) |
| 73 | \( 1 - 1.51T + 73T^{2} \) |
| 79 | \( 1 + 0.629T + 79T^{2} \) |
| 83 | \( 1 + 0.927T + 83T^{2} \) |
| 89 | \( 1 + 0.959T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.204255058520258792337200045963, −7.70365670483889523654222657939, −6.62503144005354491527480004718, −6.02317282277385711647772091525, −5.17844757133115318078840108438, −4.35746848784005618996709003532, −3.88556066268697711189721525573, −2.95520648569535150925031187865, −1.36489976196310405864722800349, −1.01805051426914283478190071118,
1.01805051426914283478190071118, 1.36489976196310405864722800349, 2.95520648569535150925031187865, 3.88556066268697711189721525573, 4.35746848784005618996709003532, 5.17844757133115318078840108438, 6.02317282277385711647772091525, 6.62503144005354491527480004718, 7.70365670483889523654222657939, 8.204255058520258792337200045963
