| L(s) = 1 | + 2.24·11-s + 5.65·17-s + 4.58·19-s − 5·25-s + 6·41-s − 1.07·43-s − 7·49-s + 5.75·59-s + 15.8·67-s − 16.9·73-s − 14.7·83-s − 5.65·89-s + 16.9·97-s + 18.2·107-s + 18·113-s + ⋯ |
| L(s) = 1 | + 0.676·11-s + 1.37·17-s + 1.05·19-s − 25-s + 0.937·41-s − 0.163·43-s − 49-s + 0.749·59-s + 1.94·67-s − 1.98·73-s − 1.61·83-s − 0.599·89-s + 1.72·97-s + 1.76·107-s + 1.69·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.155789649\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155789649\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 2.24T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 - 4.58T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 5.75T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 + 5.65T + 89T^{2} \) |
| 97 | \( 1 - 16.9T + 97T^{2} \) |
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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
−8.244811999508504807312081426048, −7.58552329396578604947500324913, −6.98290475216206760724877591999, −5.99442003471944340044173645962, −5.52434031702028917311817113157, −4.57264196491881478097447338302, −3.68997200021040974890985276774, −3.04003305886084407618584265366, −1.82743599049118109263536201005, −0.854154471349132173763761623897,
0.854154471349132173763761623897, 1.82743599049118109263536201005, 3.04003305886084407618584265366, 3.68997200021040974890985276774, 4.57264196491881478097447338302, 5.52434031702028917311817113157, 5.99442003471944340044173645962, 6.98290475216206760724877591999, 7.58552329396578604947500324913, 8.244811999508504807312081426048
