| L(s) = 1 | − 8·2-s − 27·3-s + 64·4-s − 297.·5-s + 216·6-s + 367.·7-s − 512·8-s + 729·9-s + 2.37e3·10-s − 4.73e3·11-s − 1.72e3·12-s + 450.·13-s − 2.94e3·14-s + 8.02e3·15-s + 4.09e3·16-s − 1.20e4·17-s − 5.83e3·18-s + 4.20e4·19-s − 1.90e4·20-s − 9.93e3·21-s + 3.78e4·22-s − 1.21e4·23-s + 1.38e4·24-s + 1.02e4·25-s − 3.60e3·26-s − 1.96e4·27-s + 2.35e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.06·5-s + 0.408·6-s + 0.405·7-s − 0.353·8-s + 0.333·9-s + 0.752·10-s − 1.07·11-s − 0.288·12-s + 0.0568·13-s − 0.286·14-s + 0.614·15-s + 0.250·16-s − 0.593·17-s − 0.235·18-s + 1.40·19-s − 0.531·20-s − 0.234·21-s + 0.758·22-s − 0.207·23-s + 0.204·24-s + 0.131·25-s − 0.0401·26-s − 0.192·27-s + 0.202·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 8T \) |
| 3 | \( 1 + 27T \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 5 | \( 1 + 297.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 367.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.73e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 450.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.20e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.20e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.21e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.78e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.63e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.08e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.46e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 8.84e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 6.34e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 2.56e4T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.86e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.76e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.05e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.30e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 8.15e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.72e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 3.26e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 4.06e6T + 8.07e13T^{2} \) |
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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
−9.890504906319433807996665170496, −8.822223059271659333831319519447, −7.66257629379016636229751700483, −7.45449226375021504379404527016, −5.98144778239313640133600473558, −4.95901641651460693255242964045, −3.77885724716262117344464615429, −2.41608672288672552266336355088, −0.942385457052848762709035496643, 0,
0.942385457052848762709035496643, 2.41608672288672552266336355088, 3.77885724716262117344464615429, 4.95901641651460693255242964045, 5.98144778239313640133600473558, 7.45449226375021504379404527016, 7.66257629379016636229751700483, 8.822223059271659333831319519447, 9.890504906319433807996665170496
