| L(s) = 1 | − 8·2-s − 92.6·3-s + 64·4-s − 162.·5-s + 741.·6-s − 829.·7-s − 512·8-s + 6.39e3·9-s + 1.29e3·10-s − 2.49e3·11-s − 5.93e3·12-s + 1.49e4·13-s + 6.63e3·14-s + 1.50e4·15-s + 4.09e3·16-s + 2.05e4·17-s − 5.11e4·18-s − 2.85e4·19-s − 1.03e4·20-s + 7.68e4·21-s + 1.99e4·22-s + 7.30e4·23-s + 4.74e4·24-s − 5.17e4·25-s − 1.19e5·26-s − 3.90e5·27-s − 5.31e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.98·3-s + 0.5·4-s − 0.580·5-s + 1.40·6-s − 0.914·7-s − 0.353·8-s + 2.92·9-s + 0.410·10-s − 0.564·11-s − 0.990·12-s + 1.88·13-s + 0.646·14-s + 1.15·15-s + 0.250·16-s + 1.01·17-s − 2.06·18-s − 0.955·19-s − 0.290·20-s + 1.81·21-s + 0.399·22-s + 1.25·23-s + 0.700·24-s − 0.662·25-s − 1.33·26-s − 3.81·27-s − 0.457·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 37 | \( 1 - 5.06e4T \) |
| good | 3 | \( 1 + 92.6T + 2.18e3T^{2} \) |
| 5 | \( 1 + 162.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 829.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 2.49e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.49e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.05e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.85e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.30e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 6.09e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.54e5T + 2.75e10T^{2} \) |
| 41 | \( 1 - 5.29e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.65e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.15e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.20e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.10e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 8.92e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.77e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 6.52e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.52e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.25e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.69e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.90e5T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.20e7T + 8.07e13T^{2} \) |
| 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
−12.39126822931539837696067040914, −11.06532397588645493670188159017, −10.77649359342631813209521888164, −9.426949820161535778638724918530, −7.66185920822056925424410226225, −6.46064230065258186579070048093, −5.64506388419974055478535271416, −3.85117549411986779418492347141, −1.09531673000685192307210480991, 0,
1.09531673000685192307210480991, 3.85117549411986779418492347141, 5.64506388419974055478535271416, 6.46064230065258186579070048093, 7.66185920822056925424410226225, 9.426949820161535778638724918530, 10.77649359342631813209521888164, 11.06532397588645493670188159017, 12.39126822931539837696067040914
