| L(s) = 1 | − 0.0974·2-s + 2.80·3-s − 1.99·4-s + 5-s − 0.273·6-s + 0.388·8-s + 4.89·9-s − 0.0974·10-s − 3.86·11-s − 5.59·12-s − 0.924·13-s + 2.80·15-s + 3.94·16-s − 7.27·17-s − 0.476·18-s − 4.70·19-s − 1.99·20-s + 0.376·22-s + 6.17·23-s + 1.09·24-s + 25-s + 0.0901·26-s + 5.32·27-s + 2.91·29-s − 0.273·30-s − 31-s − 1.16·32-s + ⋯ |
| L(s) = 1 | − 0.0689·2-s + 1.62·3-s − 0.995·4-s + 0.447·5-s − 0.111·6-s + 0.137·8-s + 1.63·9-s − 0.0308·10-s − 1.16·11-s − 1.61·12-s − 0.256·13-s + 0.725·15-s + 0.985·16-s − 1.76·17-s − 0.112·18-s − 1.08·19-s − 0.445·20-s + 0.0802·22-s + 1.28·23-s + 0.223·24-s + 0.200·25-s + 0.0176·26-s + 1.02·27-s + 0.541·29-s − 0.0499·30-s − 0.179·31-s − 0.205·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7595 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 + 0.0974T + 2T^{2} \) |
| 3 | \( 1 - 2.80T + 3T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 13 | \( 1 + 0.924T + 13T^{2} \) |
| 17 | \( 1 + 7.27T + 17T^{2} \) |
| 19 | \( 1 + 4.70T + 19T^{2} \) |
| 23 | \( 1 - 6.17T + 23T^{2} \) |
| 29 | \( 1 - 2.91T + 29T^{2} \) |
| 37 | \( 1 - 4.90T + 37T^{2} \) |
| 41 | \( 1 + 6.34T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 4.94T + 53T^{2} \) |
| 59 | \( 1 - 1.96T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 3.20T + 67T^{2} \) |
| 71 | \( 1 + 3.54T + 71T^{2} \) |
| 73 | \( 1 + 9.64T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 0.959T + 89T^{2} \) |
| 97 | \( 1 + 10.3T + 97T^{2} \) |
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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
−7.68481188371630507158992964745, −7.13822247255118078279151388257, −6.17541462904204215267369687837, −5.22062334429846422985715025542, −4.45329842597795495970180503999, −4.05519333744146908565875368954, −2.73264929520375419664368990014, −2.64855953202926318555338781034, −1.48212727826811638363367584310, 0,
1.48212727826811638363367584310, 2.64855953202926318555338781034, 2.73264929520375419664368990014, 4.05519333744146908565875368954, 4.45329842597795495970180503999, 5.22062334429846422985715025542, 6.17541462904204215267369687837, 7.13822247255118078279151388257, 7.68481188371630507158992964745
