| L(s) = 1 | − 32·2-s + 243·3-s + 1.02e3·4-s − 1.75e3·5-s − 7.77e3·6-s + 1.38e4·7-s − 3.27e4·8-s + 5.90e4·9-s + 5.60e4·10-s − 1.61e5·11-s + 2.48e5·12-s − 1.90e5·13-s − 4.43e5·14-s − 4.25e5·15-s + 1.04e6·16-s + 3.39e6·17-s − 1.88e6·18-s − 1.47e7·19-s − 1.79e6·20-s + 3.36e6·21-s + 5.15e6·22-s − 7.95e6·23-s − 7.96e6·24-s − 4.57e7·25-s + 6.09e6·26-s + 1.43e7·27-s + 1.41e7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.250·5-s − 0.408·6-s + 0.311·7-s − 0.353·8-s + 1/3·9-s + 0.177·10-s − 0.301·11-s + 0.288·12-s − 0.142·13-s − 0.220·14-s − 0.144·15-s + 1/4·16-s + 0.579·17-s − 0.235·18-s − 1.36·19-s − 0.125·20-s + 0.180·21-s + 0.213·22-s − 0.257·23-s − 0.204·24-s − 0.937·25-s + 0.100·26-s + 0.192·27-s + 0.155·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + p^{5} T \) |
| 3 | \( 1 - p^{5} T \) |
| 11 | \( 1 + p^{5} T \) |
| good | 5 | \( 1 + 14 p^{3} T + p^{11} T^{2} \) |
| 7 | \( 1 - 13864 T + p^{11} T^{2} \) |
| 13 | \( 1 + 190382 T + p^{11} T^{2} \) |
| 17 | \( 1 - 3395278 T + p^{11} T^{2} \) |
| 19 | \( 1 + 14773176 T + p^{11} T^{2} \) |
| 23 | \( 1 + 7959732 T + p^{11} T^{2} \) |
| 29 | \( 1 - 103198234 T + p^{11} T^{2} \) |
| 31 | \( 1 - 26907928 T + p^{11} T^{2} \) |
| 37 | \( 1 - 585244270 T + p^{11} T^{2} \) |
| 41 | \( 1 - 524400206 T + p^{11} T^{2} \) |
| 43 | \( 1 + 971194496 T + p^{11} T^{2} \) |
| 47 | \( 1 + 342773372 T + p^{11} T^{2} \) |
| 53 | \( 1 + 3132830750 T + p^{11} T^{2} \) |
| 59 | \( 1 + 4998086940 T + p^{11} T^{2} \) |
| 61 | \( 1 + 5981523902 T + p^{11} T^{2} \) |
| 67 | \( 1 + 13987592508 T + p^{11} T^{2} \) |
| 71 | \( 1 + 3744726364 T + p^{11} T^{2} \) |
| 73 | \( 1 + 23523428814 T + p^{11} T^{2} \) |
| 79 | \( 1 + 2124450104 T + p^{11} T^{2} \) |
| 83 | \( 1 + 36080767676 T + p^{11} T^{2} \) |
| 89 | \( 1 + 25240205414 T + p^{11} T^{2} \) |
| 97 | \( 1 - 19572817138 T + p^{11} T^{2} \) |
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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
−11.85545713744352435433906237331, −10.64359602780804383558090345210, −9.619695064738375788549418862501, −8.378798267365006427037524409900, −7.66029288269123235895470158554, −6.20967169690545628541359104369, −4.41843903971224641416275844180, −2.86260833773054470856456531585, −1.57449006416839453378514850927, 0,
1.57449006416839453378514850927, 2.86260833773054470856456531585, 4.41843903971224641416275844180, 6.20967169690545628541359104369, 7.66029288269123235895470158554, 8.378798267365006427037524409900, 9.619695064738375788549418862501, 10.64359602780804383558090345210, 11.85545713744352435433906237331
