| L(s) = 1 | + 0.682·2-s − 0.780·3-s − 1.53·4-s + 5-s − 0.532·6-s − 2.41·8-s − 2.39·9-s + 0.682·10-s − 4.97·11-s + 1.19·12-s + 2.11·13-s − 0.780·15-s + 1.42·16-s + 7.85·17-s − 1.63·18-s + 1.21·19-s − 1.53·20-s − 3.39·22-s − 8.53·23-s + 1.88·24-s + 25-s + 1.44·26-s + 4.20·27-s − 2.11·29-s − 0.532·30-s + 6.85·31-s + 5.79·32-s + ⋯ |
| L(s) = 1 | + 0.482·2-s − 0.450·3-s − 0.767·4-s + 0.447·5-s − 0.217·6-s − 0.852·8-s − 0.796·9-s + 0.215·10-s − 1.50·11-s + 0.345·12-s + 0.586·13-s − 0.201·15-s + 0.355·16-s + 1.90·17-s − 0.384·18-s + 0.279·19-s − 0.343·20-s − 0.723·22-s − 1.78·23-s + 0.384·24-s + 0.200·25-s + 0.283·26-s + 0.810·27-s − 0.392·29-s − 0.0973·30-s + 1.23·31-s + 1.02·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 - 0.682T + 2T^{2} \) |
| 3 | \( 1 + 0.780T + 3T^{2} \) |
| 11 | \( 1 + 4.97T + 11T^{2} \) |
| 13 | \( 1 - 2.11T + 13T^{2} \) |
| 17 | \( 1 - 7.85T + 17T^{2} \) |
| 19 | \( 1 - 1.21T + 19T^{2} \) |
| 23 | \( 1 + 8.53T + 23T^{2} \) |
| 29 | \( 1 + 2.11T + 29T^{2} \) |
| 31 | \( 1 - 6.85T + 31T^{2} \) |
| 41 | \( 1 + 1.68T + 41T^{2} \) |
| 43 | \( 1 - 5.74T + 43T^{2} \) |
| 47 | \( 1 + 7.22T + 47T^{2} \) |
| 53 | \( 1 + 2.68T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 - 8.85T + 61T^{2} \) |
| 67 | \( 1 + 5.35T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 7.15T + 73T^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 9.72T + 89T^{2} \) |
| 97 | \( 1 - 6.00T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.55613956004284543581081125134, −6.22897980924670748011645280720, −5.91736343606432317737391687592, −5.31361693912433270927040318109, −4.88317604580157700509760760183, −3.81841818210282695390459918962, −3.16116072378577891275430866581, −2.41212044970495883134132776766, −1.04385521921671105644743441153, 0,
1.04385521921671105644743441153, 2.41212044970495883134132776766, 3.16116072378577891275430866581, 3.81841818210282695390459918962, 4.88317604580157700509760760183, 5.31361693912433270927040318109, 5.91736343606432317737391687592, 6.22897980924670748011645280720, 7.55613956004284543581081125134
