| L(s) = 1 | − 42.1·2-s − 243·3-s − 273.·4-s − 6.33e3·5-s + 1.02e4·6-s − 2.29e4·7-s + 9.77e4·8-s + 5.90e4·9-s + 2.67e5·10-s − 4.34e4·11-s + 6.64e4·12-s − 8.20e5·13-s + 9.66e5·14-s + 1.54e6·15-s − 3.55e6·16-s − 5.29e6·17-s − 2.48e6·18-s − 9.56e6·19-s + 1.73e6·20-s + 5.57e6·21-s + 1.83e6·22-s − 1.56e7·23-s − 2.37e7·24-s − 8.64e6·25-s + 3.45e7·26-s − 1.43e7·27-s + 6.27e6·28-s + ⋯ |
| L(s) = 1 | − 0.930·2-s − 0.577·3-s − 0.133·4-s − 0.907·5-s + 0.537·6-s − 0.516·7-s + 1.05·8-s + 0.333·9-s + 0.844·10-s − 0.0813·11-s + 0.0770·12-s − 0.612·13-s + 0.480·14-s + 0.523·15-s − 0.848·16-s − 0.903·17-s − 0.310·18-s − 0.885·19-s + 0.121·20-s + 0.297·21-s + 0.0757·22-s − 0.506·23-s − 0.609·24-s − 0.177·25-s + 0.570·26-s − 0.192·27-s + 0.0688·28-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 | \( 1 + 42.1T + 2.04e3T^{2} \) |
| 5 | \( 1 + 6.33e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 2.29e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 4.34e4T + 2.85e11T^{2} \) |
| 13 | \( 1 + 8.20e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.29e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 9.56e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.56e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 4.39e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.36e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 4.01e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 8.87e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.42e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 3.01e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 1.28e8T + 9.26e18T^{2} \) |
| 61 | \( 1 + 9.01e8T + 4.35e19T^{2} \) |
| 67 | \( 1 + 4.70e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.04e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.87e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.04e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.18e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.60e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.89e10T + 7.15e21T^{2} \) |
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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
−10.13559634327074986162327877079, −9.191932689403741659496742061331, −8.170662102703397906913760564651, −7.33710104728801336876617976949, −6.25122914013319362300856633283, −4.72242025553331719499481936445, −3.96301640038694177597069798105, −2.25071802997293674759371326046, −0.71563742253044930249533235300, 0,
0.71563742253044930249533235300, 2.25071802997293674759371326046, 3.96301640038694177597069798105, 4.72242025553331719499481936445, 6.25122914013319362300856633283, 7.33710104728801336876617976949, 8.170662102703397906913760564651, 9.191932689403741659496742061331, 10.13559634327074986162327877079
