| L(s) = 1 | + 59.0·2-s + 243·3-s + 1.43e3·4-s + 6.91e3·5-s + 1.43e4·6-s − 1.76e4·7-s − 3.60e4·8-s + 5.90e4·9-s + 4.08e5·10-s − 6.44e5·11-s + 3.49e5·12-s + 1.28e6·13-s − 1.04e6·14-s + 1.67e6·15-s − 5.07e6·16-s − 3.91e6·17-s + 3.48e6·18-s − 1.03e7·19-s + 9.93e6·20-s − 4.29e6·21-s − 3.80e7·22-s − 1.32e7·23-s − 8.75e6·24-s − 1.04e6·25-s + 7.59e7·26-s + 1.43e7·27-s − 2.53e7·28-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.577·3-s + 0.701·4-s + 0.989·5-s + 0.753·6-s − 0.397·7-s − 0.388·8-s + 0.333·9-s + 1.29·10-s − 1.20·11-s + 0.405·12-s + 0.960·13-s − 0.518·14-s + 0.571·15-s − 1.20·16-s − 0.669·17-s + 0.434·18-s − 0.963·19-s + 0.694·20-s − 0.229·21-s − 1.57·22-s − 0.428·23-s − 0.224·24-s − 0.0214·25-s + 1.25·26-s + 0.192·27-s − 0.278·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 + 7.14e8T \) |
| good | 2 | \( 1 - 59.0T + 2.04e3T^{2} \) |
| 5 | \( 1 - 6.91e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 1.76e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 6.44e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.28e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.91e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.03e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.32e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 5.58e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.37e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.25e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.19e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.72e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.10e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 4.23e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 9.63e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.37e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.14e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.75e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.33e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.20e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.88e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 9.04e10T + 7.15e21T^{2} \) |
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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
−10.21256845240585210656623054007, −9.187365242881612574354530438640, −8.186092149156094894886047223587, −6.61380374215170487126603373479, −5.89617783127776153390205663304, −4.86371043469076479190536945791, −3.74751055265794184899338656182, −2.71952771839357254152418427827, −1.87891971505825064745244267100, 0,
1.87891971505825064745244267100, 2.71952771839357254152418427827, 3.74751055265794184899338656182, 4.86371043469076479190536945791, 5.89617783127776153390205663304, 6.61380374215170487126603373479, 8.186092149156094894886047223587, 9.187365242881612574354530438640, 10.21256845240585210656623054007
