L(s) = 1 | − 102.·2-s + 729·3-s + 2.27e3·4-s − 9.86e3·5-s − 7.45e4·6-s + 4.39e5·7-s + 6.05e5·8-s + 5.31e5·9-s + 1.00e6·10-s − 9.59e5·11-s + 1.65e6·12-s + 5.60e6·13-s − 4.50e7·14-s − 7.18e6·15-s − 8.05e7·16-s − 1.88e8·17-s − 5.43e7·18-s − 3.80e6·19-s − 2.24e7·20-s + 3.20e8·21-s + 9.81e7·22-s + 1.27e9·23-s + 4.41e8·24-s − 1.12e9·25-s − 5.73e8·26-s + 3.87e8·27-s + 1.00e9·28-s + ⋯ |
L(s) = 1 | − 1.13·2-s + 0.577·3-s + 0.277·4-s − 0.282·5-s − 0.652·6-s + 1.41·7-s + 0.816·8-s + 0.333·9-s + 0.319·10-s − 0.163·11-s + 0.160·12-s + 0.322·13-s − 1.59·14-s − 0.162·15-s − 1.20·16-s − 1.89·17-s − 0.376·18-s − 0.0185·19-s − 0.0784·20-s + 0.815·21-s + 0.184·22-s + 1.79·23-s + 0.471·24-s − 0.920·25-s − 0.364·26-s + 0.192·27-s + 0.392·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(\approx\) |
\(1.662909240\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.662909240\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 + 4.21e10T \) |
good | 2 | \( 1 + 102.T + 8.19e3T^{2} \) |
| 5 | \( 1 + 9.86e3T + 1.22e9T^{2} \) |
| 7 | \( 1 - 4.39e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 9.59e5T + 3.45e13T^{2} \) |
| 13 | \( 1 - 5.60e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.88e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 3.80e6T + 4.20e16T^{2} \) |
| 23 | \( 1 - 1.27e9T + 5.04e17T^{2} \) |
| 29 | \( 1 - 4.95e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 8.24e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 2.50e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 3.30e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 4.90e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 8.62e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.96e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 6.27e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 6.94e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.94e11T + 1.16e24T^{2} \) |
| 73 | \( 1 - 3.28e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 3.01e10T + 4.66e24T^{2} \) |
| 83 | \( 1 - 5.53e11T + 8.87e24T^{2} \) |
| 89 | \( 1 + 2.05e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 5.32e12T + 6.73e25T^{2} \) |
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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
−10.33827451939389125020607118968, −8.842314882884537026471713591943, −8.665953832507744198698594589416, −7.69388366159862837021375085650, −6.78937368074113907489959939478, −4.90831779321139844549419791132, −4.25446340533451014634715702369, −2.56705003412517147765057149546, −1.57671998613038526132747857337, −0.67257818921093984872470953401,
0.67257818921093984872470953401, 1.57671998613038526132747857337, 2.56705003412517147765057149546, 4.25446340533451014634715702369, 4.90831779321139844549419791132, 6.78937368074113907489959939478, 7.69388366159862837021375085650, 8.665953832507744198698594589416, 8.842314882884537026471713591943, 10.33827451939389125020607118968