| L(s) = 1 | − 28.8·2-s − 81·3-s + 321.·4-s − 2.14e3·5-s + 2.33e3·6-s + 2.49e3·7-s + 5.49e3·8-s + 6.56e3·9-s + 6.19e4·10-s − 2.48e4·11-s − 2.60e4·12-s − 1.27e5·13-s − 7.20e4·14-s + 1.73e5·15-s − 3.23e5·16-s − 4.24e5·17-s − 1.89e5·18-s − 3.08e5·19-s − 6.90e5·20-s − 2.02e5·21-s + 7.18e5·22-s + 1.81e6·23-s − 4.44e5·24-s + 2.64e6·25-s + 3.68e6·26-s − 5.31e5·27-s + 8.03e5·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.577·3-s + 0.628·4-s − 1.53·5-s + 0.736·6-s + 0.392·7-s + 0.474·8-s + 0.333·9-s + 1.95·10-s − 0.512·11-s − 0.362·12-s − 1.23·13-s − 0.501·14-s + 0.886·15-s − 1.23·16-s − 1.23·17-s − 0.425·18-s − 0.543·19-s − 0.964·20-s − 0.226·21-s + 0.653·22-s + 1.35·23-s − 0.273·24-s + 1.35·25-s + 1.57·26-s − 0.192·27-s + 0.246·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 + 28.8T + 512T^{2} \) |
| 5 | \( 1 + 2.14e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 2.49e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.48e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.27e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.24e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 3.08e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.81e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.84e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 9.38e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 4.96e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.21e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.79e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.69e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 8.66e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 7.59e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.33e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.85e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.90e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.35e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.69e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 6.99e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.03e9T + 7.60e17T^{2} \) |
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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
−10.79423124825681637499300405471, −9.381598958144489337043845543594, −8.456342816956697861137593151497, −7.53056763982142647099429898546, −6.96251057551360552887861515094, −4.99114331996046620432235985665, −4.19421514974768286947684858475, −2.36309267469576617236419942022, −0.75510310353169709631143697861, 0,
0.75510310353169709631143697861, 2.36309267469576617236419942022, 4.19421514974768286947684858475, 4.99114331996046620432235985665, 6.96251057551360552887861515094, 7.53056763982142647099429898546, 8.456342816956697861137593151497, 9.381598958144489337043845543594, 10.79423124825681637499300405471
