L(s) = 1 | − 64.1·2-s − 243·3-s + 2.07e3·4-s − 6.30e3·5-s + 1.55e4·6-s + 8.46e4·7-s − 1.56e3·8-s + 5.90e4·9-s + 4.04e5·10-s − 9.02e5·11-s − 5.03e5·12-s − 1.37e6·13-s − 5.43e6·14-s + 1.53e6·15-s − 4.14e6·16-s + 6.10e6·17-s − 3.79e6·18-s + 1.53e7·19-s − 1.30e7·20-s − 2.05e7·21-s + 5.79e7·22-s + 5.45e7·23-s + 3.79e5·24-s − 9.04e6·25-s + 8.82e7·26-s − 1.43e7·27-s + 1.75e8·28-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 1.01·4-s − 0.902·5-s + 0.818·6-s + 1.90·7-s − 0.0168·8-s + 0.333·9-s + 1.28·10-s − 1.69·11-s − 0.584·12-s − 1.02·13-s − 2.69·14-s + 0.521·15-s − 0.987·16-s + 1.04·17-s − 0.472·18-s + 1.42·19-s − 0.913·20-s − 1.09·21-s + 2.39·22-s + 1.76·23-s + 0.00972·24-s − 0.185·25-s + 1.45·26-s − 0.192·27-s + 1.92·28-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
\(L(6)\) |
\(\approx\) |
\(0.7530534544\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7530534544\) |
\(L(\frac{13}{2})\) |
|
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 | \( 1 + 64.1T + 2.04e3T^{2} \) |
| 5 | \( 1 + 6.30e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 8.46e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 9.02e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.37e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 6.10e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.53e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 5.45e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 3.20e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.01e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 6.20e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 1.18e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 2.91e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 6.14e7T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.65e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 1.13e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 8.11e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.75e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.78e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.32e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 8.67e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 7.88e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 6.50e10T + 7.15e21T^{2} \) |
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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
−10.60033212285896145576687428489, −9.802390545496031336247741899772, −8.352736205860585406943474666810, −7.72727084770310313835266508767, −7.35277688667361182567706397497, −5.16618095301261905976851265131, −4.76864622466574981561009029470, −2.73634732121980471693027839923, −1.34712815620250782231852926704, −0.56932938228978206196610802612,
0.56932938228978206196610802612, 1.34712815620250782231852926704, 2.73634732121980471693027839923, 4.76864622466574981561009029470, 5.16618095301261905976851265131, 7.35277688667361182567706397497, 7.72727084770310313835266508767, 8.352736205860585406943474666810, 9.802390545496031336247741899772, 10.60033212285896145576687428489