L(s) = 1 | − 2-s − 4-s + 2·5-s + 7-s + 3·8-s − 3·9-s − 2·10-s + 4·11-s + 6·13-s − 14-s − 16-s − 2·17-s + 3·18-s + 4·19-s − 2·20-s − 4·22-s − 23-s − 25-s − 6·26-s − 28-s − 2·29-s − 4·31-s − 5·32-s + 2·34-s + 2·35-s + 3·36-s − 2·37-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 0.377·7-s + 1.06·8-s − 9-s − 0.632·10-s + 1.20·11-s + 1.66·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.707·18-s + 0.917·19-s − 0.447·20-s − 0.852·22-s − 0.208·23-s − 1/5·25-s − 1.17·26-s − 0.188·28-s − 0.371·29-s − 0.718·31-s − 0.883·32-s + 0.342·34-s + 0.338·35-s + 1/2·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8628343267\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8628343267\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.12082677402822691324891147054, −11.57306814136850016603459846709, −10.82999474885334929942270674109, −9.519763597988292185167511678467, −8.955336427994798867614764820369, −8.048209611049889306918613094768, −6.43402806030236505610147737889, −5.38200398568328423768601134305, −3.76496444511642110376256706933, −1.51926854854030366908096170303,
1.51926854854030366908096170303, 3.76496444511642110376256706933, 5.38200398568328423768601134305, 6.43402806030236505610147737889, 8.048209611049889306918613094768, 8.955336427994798867614764820369, 9.519763597988292185167511678467, 10.82999474885334929942270674109, 11.57306814136850016603459846709, 13.12082677402822691324891147054