| L(s) = 1 | − 169.·2-s − 729·3-s + 2.03e4·4-s + 1.50e4·5-s + 1.23e5·6-s − 2.57e5·7-s − 2.06e6·8-s + 5.31e5·9-s − 2.54e6·10-s − 4.81e6·11-s − 1.48e7·12-s − 3.34e7·13-s + 4.35e7·14-s − 1.09e7·15-s + 1.81e8·16-s − 1.05e8·17-s − 8.98e7·18-s + 1.32e8·19-s + 3.06e8·20-s + 1.87e8·21-s + 8.13e8·22-s + 7.80e8·23-s + 1.50e9·24-s − 9.94e8·25-s + 5.66e9·26-s − 3.87e8·27-s − 5.25e9·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s − 0.577·3-s + 2.48·4-s + 0.430·5-s + 1.07·6-s − 0.826·7-s − 2.78·8-s + 0.333·9-s − 0.803·10-s − 0.819·11-s − 1.43·12-s − 1.92·13-s + 1.54·14-s − 0.248·15-s + 2.70·16-s − 1.06·17-s − 0.622·18-s + 0.646·19-s + 1.07·20-s + 0.477·21-s + 1.53·22-s + 1.09·23-s + 1.60·24-s − 0.814·25-s + 3.59·26-s − 0.192·27-s − 2.05·28-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 | \( 1 + 169.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 1.50e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 2.57e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 4.81e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 3.34e7T + 3.02e14T^{2} \) |
| 17 | \( 1 + 1.05e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 1.32e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 7.80e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 9.46e8T + 1.02e19T^{2} \) |
| 31 | \( 1 - 7.66e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.25e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.31e10T + 9.25e20T^{2} \) |
| 43 | \( 1 - 2.54e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 9.37e9T + 5.46e21T^{2} \) |
| 53 | \( 1 - 1.13e11T + 2.60e22T^{2} \) |
| 61 | \( 1 + 1.44e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 4.28e11T + 5.48e23T^{2} \) |
| 71 | \( 1 - 1.96e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.02e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 7.60e11T + 4.66e24T^{2} \) |
| 83 | \( 1 - 1.79e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 8.56e12T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.76e12T + 6.73e25T^{2} \) |
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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
−9.808867060922721098769630364897, −9.179319971378387491470413223141, −7.82962808747132586691243678991, −7.06571697294410024055227162372, −6.22861708446718597567196205642, −4.97484125373384975026052213632, −2.83020384061588135075823955406, −2.13936529257741338964172842554, −0.73579023098383782971654199549, 0,
0.73579023098383782971654199549, 2.13936529257741338964172842554, 2.83020384061588135075823955406, 4.97484125373384975026052213632, 6.22861708446718597567196205642, 7.06571697294410024055227162372, 7.82962808747132586691243678991, 9.179319971378387491470413223141, 9.808867060922721098769630364897
