| L(s) = 1 | − 1.53·2-s − 0.347·3-s + 0.347·4-s + 0.532·6-s + 0.347·7-s + 2.53·8-s − 2.87·9-s − 1.34·11-s − 0.120·12-s + 3.41·13-s − 0.532·14-s − 4.57·16-s + 0.879·17-s + 4.41·18-s − 3.53·19-s − 0.120·21-s + 2.06·22-s + 1.53·23-s − 0.879·24-s − 5.22·26-s + 2.04·27-s + 0.120·28-s − 0.120·29-s + 3.82·31-s + 1.94·32-s + 0.467·33-s − 1.34·34-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 0.200·3-s + 0.173·4-s + 0.217·6-s + 0.131·7-s + 0.895·8-s − 0.959·9-s − 0.406·11-s − 0.0348·12-s + 0.946·13-s − 0.142·14-s − 1.14·16-s + 0.213·17-s + 1.03·18-s − 0.810·19-s − 0.0263·21-s + 0.440·22-s + 0.319·23-s − 0.179·24-s − 1.02·26-s + 0.392·27-s + 0.0227·28-s − 0.0223·29-s + 0.686·31-s + 0.343·32-s + 0.0814·33-s − 0.231·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 + 1.53T + 2T^{2} \) |
| 3 | \( 1 + 0.347T + 3T^{2} \) |
| 7 | \( 1 - 0.347T + 7T^{2} \) |
| 11 | \( 1 + 1.34T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 - 0.879T + 17T^{2} \) |
| 19 | \( 1 + 3.53T + 19T^{2} \) |
| 23 | \( 1 - 1.53T + 23T^{2} \) |
| 29 | \( 1 + 0.120T + 29T^{2} \) |
| 31 | \( 1 - 3.82T + 31T^{2} \) |
| 37 | \( 1 + 8.17T + 37T^{2} \) |
| 41 | \( 1 - 1.98T + 41T^{2} \) |
| 43 | \( 1 - 3.41T + 43T^{2} \) |
| 53 | \( 1 - 2.81T + 53T^{2} \) |
| 59 | \( 1 - 2.63T + 59T^{2} \) |
| 61 | \( 1 + 6.69T + 61T^{2} \) |
| 67 | \( 1 + 6.33T + 67T^{2} \) |
| 71 | \( 1 + 9.66T + 71T^{2} \) |
| 73 | \( 1 - 2.04T + 73T^{2} \) |
| 79 | \( 1 + 9.94T + 79T^{2} \) |
| 83 | \( 1 + 4.87T + 83T^{2} \) |
| 89 | \( 1 + 8.55T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
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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
−9.197987416333618375157207945018, −8.539481156724479578255194576283, −8.090587164499707583808526557867, −7.05493423173284676045093790404, −6.09128287986822522501778870132, −5.17380418778063357547261951637, −4.12365015024383656270491922227, −2.82235433420766584674437859171, −1.43404151912888000355760499212, 0,
1.43404151912888000355760499212, 2.82235433420766584674437859171, 4.12365015024383656270491922227, 5.17380418778063357547261951637, 6.09128287986822522501778870132, 7.05493423173284676045093790404, 8.090587164499707583808526557867, 8.539481156724479578255194576283, 9.197987416333618375157207945018
