| L(s) = 1 | + 32.2·2-s + 243·3-s − 1.01e3·4-s − 5.50e3·5-s + 7.82e3·6-s − 3.67e4·7-s − 9.85e4·8-s + 5.90e4·9-s − 1.77e5·10-s + 7.85e5·11-s − 2.45e5·12-s + 1.64e6·13-s − 1.18e6·14-s − 1.33e6·15-s − 1.10e6·16-s + 1.42e4·17-s + 1.90e6·18-s − 8.87e6·19-s + 5.56e6·20-s − 8.92e6·21-s + 2.52e7·22-s + 2.97e7·23-s − 2.39e7·24-s − 1.84e7·25-s + 5.30e7·26-s + 1.43e7·27-s + 3.71e7·28-s + ⋯ |
| L(s) = 1 | + 0.711·2-s + 0.577·3-s − 0.493·4-s − 0.788·5-s + 0.410·6-s − 0.826·7-s − 1.06·8-s + 0.333·9-s − 0.561·10-s + 1.47·11-s − 0.284·12-s + 1.23·13-s − 0.587·14-s − 0.455·15-s − 0.262·16-s + 0.00243·17-s + 0.237·18-s − 0.822·19-s + 0.389·20-s − 0.476·21-s + 1.04·22-s + 0.962·23-s − 0.613·24-s − 0.378·25-s + 0.876·26-s + 0.192·27-s + 0.407·28-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 | \( 1 - 32.2T + 2.04e3T^{2} \) |
| 5 | \( 1 + 5.50e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 3.67e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 7.85e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.64e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.42e4T + 3.42e13T^{2} \) |
| 19 | \( 1 + 8.87e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.97e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 6.76e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 6.05e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.57e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.38e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.09e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.74e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.03e8T + 9.26e18T^{2} \) |
| 61 | \( 1 - 1.62e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.24e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 8.29e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.93e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.88e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.23e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 9.79e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 2.14e10T + 7.15e21T^{2} \) |
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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
−10.02361637493806264929695154500, −8.926003899501891578085303199819, −8.458178838314239892160429784280, −6.86804823996452912113752857055, −6.07170315582238683210131973363, −4.48321832051747685131866754425, −3.75114317752071535073002522266, −3.09986023936051608684577497690, −1.27248707120639021321328990185, 0,
1.27248707120639021321328990185, 3.09986023936051608684577497690, 3.75114317752071535073002522266, 4.48321832051747685131866754425, 6.07170315582238683210131973363, 6.86804823996452912113752857055, 8.458178838314239892160429784280, 8.926003899501891578085303199819, 10.02361637493806264929695154500
