| L(s) = 1 | + 8.42·2-s + 9·3-s + 38.9·4-s − 30.3·5-s + 75.8·6-s − 197.·7-s + 58.7·8-s + 81·9-s − 255.·10-s − 604.·11-s + 350.·12-s − 893.·13-s − 1.66e3·14-s − 273.·15-s − 752.·16-s + 1.08e3·17-s + 682.·18-s + 2.02e3·19-s − 1.18e3·20-s − 1.77e3·21-s − 5.09e3·22-s + 1.32e3·23-s + 528.·24-s − 2.20e3·25-s − 7.52e3·26-s + 729·27-s − 7.70e3·28-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 0.577·3-s + 1.21·4-s − 0.543·5-s + 0.859·6-s − 1.52·7-s + 0.324·8-s + 0.333·9-s − 0.809·10-s − 1.50·11-s + 0.703·12-s − 1.46·13-s − 2.27·14-s − 0.313·15-s − 0.734·16-s + 0.913·17-s + 0.496·18-s + 1.28·19-s − 0.661·20-s − 0.880·21-s − 2.24·22-s + 0.521·23-s + 0.187·24-s − 0.704·25-s − 2.18·26-s + 0.192·27-s − 1.85·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 - 8.42T + 32T^{2} \) |
| 5 | \( 1 + 30.3T + 3.12e3T^{2} \) |
| 7 | \( 1 + 197.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 604.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 893.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.08e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.02e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.32e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.78e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.30e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.06e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.39e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.34e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.74e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 1.17e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.69e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.21e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.14e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.01e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.82e5T + 8.58e9T^{2} \) |
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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
−11.92132643091412496622430194535, −10.25946091780862039779177289654, −9.509307535355125826199026492451, −7.84655299565924140550587119813, −6.98820985084515780331575077900, −5.64762525665602448963489283050, −4.63806443404185257862366316040, −3.17362033918638656828068589940, −2.82636828125798405848457810768, 0,
2.82636828125798405848457810768, 3.17362033918638656828068589940, 4.63806443404185257862366316040, 5.64762525665602448963489283050, 6.98820985084515780331575077900, 7.84655299565924140550587119813, 9.509307535355125826199026492451, 10.25946091780862039779177289654, 11.92132643091412496622430194535
