| L(s) = 1 | − 153.·2-s + 729·3-s + 1.52e4·4-s + 5.84e4·5-s − 1.11e5·6-s + 5.70e4·7-s − 1.08e6·8-s + 5.31e5·9-s − 8.95e6·10-s − 8.87e6·11-s + 1.11e7·12-s − 4.66e6·13-s − 8.73e6·14-s + 4.25e7·15-s + 4.12e7·16-s − 9.15e7·17-s − 8.14e7·18-s − 1.77e8·19-s + 8.92e8·20-s + 4.15e7·21-s + 1.35e9·22-s + 7.81e8·23-s − 7.91e8·24-s + 2.19e9·25-s + 7.14e8·26-s + 3.87e8·27-s + 8.71e8·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 0.577·3-s + 1.86·4-s + 1.67·5-s − 0.977·6-s + 0.183·7-s − 1.46·8-s + 0.333·9-s − 2.83·10-s − 1.50·11-s + 1.07·12-s − 0.267·13-s − 0.310·14-s + 0.965·15-s + 0.614·16-s − 0.919·17-s − 0.564·18-s − 0.864·19-s + 3.11·20-s + 0.105·21-s + 2.55·22-s + 1.10·23-s − 0.845·24-s + 1.79·25-s + 0.453·26-s + 0.192·27-s + 0.341·28-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
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + 153.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 5.84e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 5.70e4T + 9.68e10T^{2} \) |
| 11 | \( 1 + 8.87e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 4.66e6T + 3.02e14T^{2} \) |
| 17 | \( 1 + 9.15e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + 1.77e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 7.81e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 3.99e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 5.30e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.42e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.50e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 2.93e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 4.10e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.99e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 6.54e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.22e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.52e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.65e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 2.59e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.52e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 6.85e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 5.66e12T + 6.73e25T^{2} \) |
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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
−9.707496004924245265948882207485, −8.984667568142692525646496364674, −8.154065058148616411841200192239, −7.13222401460446291250088401856, −6.12038578559237782093496052695, −4.87118454414644309378069538169, −2.60781716758110792112805645529, −2.25934008696668939153179831636, −1.24330717681974243454040292261, 0,
1.24330717681974243454040292261, 2.25934008696668939153179831636, 2.60781716758110792112805645529, 4.87118454414644309378069538169, 6.12038578559237782093496052695, 7.13222401460446291250088401856, 8.154065058148616411841200192239, 8.984667568142692525646496364674, 9.707496004924245265948882207485
