| L(s) = 1 | + 26.5·2-s − 204.·3-s + 194.·4-s − 2.37e3·5-s − 5.44e3·6-s + 6.24e3·7-s − 8.43e3·8-s + 2.23e4·9-s − 6.30e4·10-s − 8.26e4·11-s − 3.98e4·12-s + 1.10e5·13-s + 1.65e5·14-s + 4.86e5·15-s − 3.23e5·16-s + 5.93e5·18-s + 5.62e5·19-s − 4.61e5·20-s − 1.27e6·21-s − 2.19e6·22-s − 1.19e6·23-s + 1.72e6·24-s + 3.67e6·25-s + 2.94e6·26-s − 5.40e5·27-s + 1.21e6·28-s + 1.98e6·29-s + ⋯ |
| L(s) = 1 | + 1.17·2-s − 1.46·3-s + 0.379·4-s − 1.69·5-s − 1.71·6-s + 0.982·7-s − 0.728·8-s + 1.13·9-s − 1.99·10-s − 1.70·11-s − 0.555·12-s + 1.07·13-s + 1.15·14-s + 2.47·15-s − 1.23·16-s + 1.33·18-s + 0.989·19-s − 0.644·20-s − 1.43·21-s − 1.99·22-s − 0.890·23-s + 1.06·24-s + 1.88·25-s + 1.26·26-s − 0.195·27-s + 0.373·28-s + 0.521·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 - 26.5T + 512T^{2} \) |
| 3 | \( 1 + 204.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.37e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 6.24e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 8.26e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.10e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 5.62e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.19e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.98e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 9.43e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.84e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.19e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.42e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.11e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.19e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.21e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.01e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.11e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.37e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.34e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.77e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.29e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.82e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.77e7T + 7.60e17T^{2} \) |
| 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
−10.33678215641007014960018856379, −8.368868250382678458247861896321, −7.77338192838365187752773642342, −6.48485681150258261892662822650, −5.42514685710725072995948442689, −4.82531224251414892304257716570, −4.08603734036040459208355532506, −2.92336098977872768297774769859, −0.893254292262306518877130073058, 0,
0.893254292262306518877130073058, 2.92336098977872768297774769859, 4.08603734036040459208355532506, 4.82531224251414892304257716570, 5.42514685710725072995948442689, 6.48485681150258261892662822650, 7.77338192838365187752773642342, 8.368868250382678458247861896321, 10.33678215641007014960018856379
