L(s) = 1 | − 2-s + 4-s − 1.41·5-s − 1.27·7-s − 8-s + 1.41·10-s − 5.29·11-s + 6.20·13-s + 1.27·14-s + 16-s − 6.19·17-s + 4.29·19-s − 1.41·20-s + 5.29·22-s + 6.65·23-s − 2.99·25-s − 6.20·26-s − 1.27·28-s − 4.65·29-s + 0.972·31-s − 32-s + 6.19·34-s + 1.80·35-s + 0.986·37-s − 4.29·38-s + 1.41·40-s + 8.34·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.633·5-s − 0.482·7-s − 0.353·8-s + 0.448·10-s − 1.59·11-s + 1.72·13-s + 0.340·14-s + 0.250·16-s − 1.50·17-s + 0.984·19-s − 0.316·20-s + 1.12·22-s + 1.38·23-s − 0.598·25-s − 1.21·26-s − 0.241·28-s − 0.864·29-s + 0.174·31-s − 0.176·32-s + 1.06·34-s + 0.305·35-s + 0.162·37-s − 0.696·38-s + 0.224·40-s + 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 1.27T + 7T^{2} \) |
| 11 | \( 1 + 5.29T + 11T^{2} \) |
| 13 | \( 1 - 6.20T + 13T^{2} \) |
| 17 | \( 1 + 6.19T + 17T^{2} \) |
| 19 | \( 1 - 4.29T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 - 0.972T + 31T^{2} \) |
| 37 | \( 1 - 0.986T + 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 - 3.52T + 43T^{2} \) |
| 47 | \( 1 + 4.00T + 47T^{2} \) |
| 53 | \( 1 - 0.663T + 53T^{2} \) |
| 59 | \( 1 - 4.27T + 59T^{2} \) |
| 61 | \( 1 - 3.90T + 61T^{2} \) |
| 67 | \( 1 - 2.70T + 67T^{2} \) |
| 71 | \( 1 - 2.68T + 71T^{2} \) |
| 73 | \( 1 + 4.49T + 73T^{2} \) |
| 79 | \( 1 + 5.09T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 8.01T + 89T^{2} \) |
| 97 | \( 1 + 2.43T + 97T^{2} \) |
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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
−7.56294995532587609001734253875, −7.00045828638953358656030654718, −6.17183818926506125337250618459, −5.55917505571155152290579326557, −4.65305719581592870086408882928, −3.71947337497233594365799831246, −3.05239163515661845728978271170, −2.22906556444728703537734842079, −1.01297343551217573691322013387, 0,
1.01297343551217573691322013387, 2.22906556444728703537734842079, 3.05239163515661845728978271170, 3.71947337497233594365799831246, 4.65305719581592870086408882928, 5.55917505571155152290579326557, 6.17183818926506125337250618459, 7.00045828638953358656030654718, 7.56294995532587609001734253875