L(s) = 1 | + 2.34·2-s − 0.160·3-s + 3.49·4-s + 5-s − 0.376·6-s + 7-s + 3.49·8-s − 2.97·9-s + 2.34·10-s − 5.21·11-s − 0.561·12-s − 2.95·13-s + 2.34·14-s − 0.160·15-s + 1.20·16-s + 6.94·17-s − 6.96·18-s − 7.34·19-s + 3.49·20-s − 0.160·21-s − 12.2·22-s + 6.50·23-s − 0.562·24-s + 25-s − 6.93·26-s + 0.960·27-s + 3.49·28-s + ⋯ |
L(s) = 1 | + 1.65·2-s − 0.0928·3-s + 1.74·4-s + 0.447·5-s − 0.153·6-s + 0.377·7-s + 1.23·8-s − 0.991·9-s + 0.741·10-s − 1.57·11-s − 0.162·12-s − 0.820·13-s + 0.626·14-s − 0.0415·15-s + 0.302·16-s + 1.68·17-s − 1.64·18-s − 1.68·19-s + 0.780·20-s − 0.0350·21-s − 2.60·22-s + 1.35·23-s − 0.114·24-s + 0.200·25-s − 1.36·26-s + 0.184·27-s + 0.659·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 - 2.34T + 2T^{2} \) |
| 3 | \( 1 + 0.160T + 3T^{2} \) |
| 11 | \( 1 + 5.21T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 - 6.94T + 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 23 | \( 1 - 6.50T + 23T^{2} \) |
| 29 | \( 1 + 0.552T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 + 7.29T + 37T^{2} \) |
| 41 | \( 1 + 5.04T + 41T^{2} \) |
| 43 | \( 1 - 6.02T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 1.44T + 53T^{2} \) |
| 59 | \( 1 - 9.35T + 59T^{2} \) |
| 61 | \( 1 - 1.27T + 61T^{2} \) |
| 67 | \( 1 - 2.75T + 67T^{2} \) |
| 71 | \( 1 - 4.37T + 71T^{2} \) |
| 73 | \( 1 + 1.97T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 8.55T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + 8.98T + 97T^{2} \) |
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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
−7.20086913085047516506007908729, −6.62453849409529550322403236057, −5.63508293653143796657351469506, −5.34945526621302941991973630211, −4.96613798069918358087318128317, −3.99152024428731150607373790803, −2.96657669592908838543730699808, −2.70678316766015581085074795777, −1.75115656451243411518442821678, 0,
1.75115656451243411518442821678, 2.70678316766015581085074795777, 2.96657669592908838543730699808, 3.99152024428731150607373790803, 4.96613798069918358087318128317, 5.34945526621302941991973630211, 5.63508293653143796657351469506, 6.62453849409529550322403236057, 7.20086913085047516506007908729