| L(s) = 1 | + 8·2-s − 44.7·3-s + 64·4-s − 46.8·5-s − 357.·6-s − 343·7-s + 512·8-s − 187.·9-s − 374.·10-s + 7.41e3·11-s − 2.86e3·12-s − 2.19e3·13-s − 2.74e3·14-s + 2.09e3·15-s + 4.09e3·16-s − 1.17e4·17-s − 1.49e3·18-s + 3.78e4·19-s − 2.99e3·20-s + 1.53e4·21-s + 5.93e4·22-s − 4.62e3·23-s − 2.28e4·24-s − 7.59e4·25-s − 1.75e4·26-s + 1.06e5·27-s − 2.19e4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.956·3-s + 0.5·4-s − 0.167·5-s − 0.676·6-s − 0.377·7-s + 0.353·8-s − 0.0855·9-s − 0.118·10-s + 1.67·11-s − 0.478·12-s − 0.277·13-s − 0.267·14-s + 0.160·15-s + 0.250·16-s − 0.579·17-s − 0.0604·18-s + 1.26·19-s − 0.0837·20-s + 0.361·21-s + 1.18·22-s − 0.0792·23-s − 0.338·24-s − 0.971·25-s − 0.196·26-s + 1.03·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 8T \) |
| 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 3 | \( 1 + 44.7T + 2.18e3T^{2} \) |
| 5 | \( 1 + 46.8T + 7.81e4T^{2} \) |
| 11 | \( 1 - 7.41e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.17e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.78e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.62e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.47e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 7.08e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.34e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.81e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.32e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 2.90e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.71e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.39e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.31e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.09e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.34e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.48e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.47e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.74e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.55e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 3.44e6T + 8.07e13T^{2} \) |
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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
−11.39974399115241433116101889782, −10.01644107204152604978391482550, −8.944601932809223119112670553153, −7.33002626429302553046383607616, −6.39764210383932576193678897987, −5.58483105479448335724630362185, −4.37487355214088104207185141240, −3.25630789630917124445164591837, −1.49230760558791666162544992263, 0,
1.49230760558791666162544992263, 3.25630789630917124445164591837, 4.37487355214088104207185141240, 5.58483105479448335724630362185, 6.39764210383932576193678897987, 7.33002626429302553046383607616, 8.944601932809223119112670553153, 10.01644107204152604978391482550, 11.39974399115241433116101889782
