| L(s) = 1 | − 86.7·2-s + 243·3-s + 5.47e3·4-s − 1.17e4·5-s − 2.10e4·6-s + 2.42e4·7-s − 2.97e5·8-s + 5.90e4·9-s + 1.01e6·10-s − 2.25e5·11-s + 1.33e6·12-s − 1.93e6·13-s − 2.10e6·14-s − 2.84e6·15-s + 1.45e7·16-s − 7.58e6·17-s − 5.12e6·18-s + 2.02e7·19-s − 6.42e7·20-s + 5.88e6·21-s + 1.95e7·22-s + 4.13e6·23-s − 7.22e7·24-s + 8.86e7·25-s + 1.67e8·26-s + 1.43e7·27-s + 1.32e8·28-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 0.577·3-s + 2.67·4-s − 1.67·5-s − 1.10·6-s + 0.545·7-s − 3.20·8-s + 0.333·9-s + 3.21·10-s − 0.422·11-s + 1.54·12-s − 1.44·13-s − 1.04·14-s − 0.968·15-s + 3.47·16-s − 1.29·17-s − 0.638·18-s + 1.87·19-s − 4.48·20-s + 0.314·21-s + 0.809·22-s + 0.133·23-s − 1.85·24-s + 1.81·25-s + 2.77·26-s + 0.192·27-s + 1.45·28-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 | \( 1 + 86.7T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.17e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.42e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 2.25e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.93e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 7.58e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 2.02e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.13e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.28e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.26e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 1.58e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 7.42e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.64e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.37e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.06e8T + 9.26e18T^{2} \) |
| 61 | \( 1 - 1.25e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 3.89e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.09e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.32e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.04e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.10e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 6.05e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.06e11T + 7.15e21T^{2} \) |
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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
−9.921883749467526801760611030172, −9.025869057921686846286279202838, −8.064544655975542191067449896984, −7.60024715538521404431294861409, −6.91123207399527314966720033863, −4.80855938148107968153974762673, −3.22126171104857069657859216125, −2.27434734708961379901365630836, −0.896759049382490569846956248121, 0,
0.896759049382490569846956248121, 2.27434734708961379901365630836, 3.22126171104857069657859216125, 4.80855938148107968153974762673, 6.91123207399527314966720033863, 7.60024715538521404431294861409, 8.064544655975542191067449896984, 9.025869057921686846286279202838, 9.921883749467526801760611030172
