| L(s) = 1 | + 13.5·2-s + 243·3-s − 1.86e3·4-s − 1.20e4·5-s + 3.28e3·6-s − 1.96e4·7-s − 5.29e4·8-s + 5.90e4·9-s − 1.62e5·10-s + 2.75e5·11-s − 4.53e5·12-s + 2.60e5·13-s − 2.66e5·14-s − 2.91e6·15-s + 3.10e6·16-s + 2.78e6·17-s + 7.98e5·18-s + 1.56e7·19-s + 2.23e7·20-s − 4.78e6·21-s + 3.72e6·22-s − 5.51e7·23-s − 1.28e7·24-s + 9.52e7·25-s + 3.52e6·26-s + 1.43e7·27-s + 3.67e7·28-s + ⋯ |
| L(s) = 1 | + 0.298·2-s + 0.577·3-s − 0.910·4-s − 1.71·5-s + 0.172·6-s − 0.442·7-s − 0.570·8-s + 0.333·9-s − 0.513·10-s + 0.516·11-s − 0.525·12-s + 0.194·13-s − 0.132·14-s − 0.991·15-s + 0.740·16-s + 0.475·17-s + 0.0995·18-s + 1.45·19-s + 1.56·20-s − 0.255·21-s + 0.154·22-s − 1.78·23-s − 0.329·24-s + 1.95·25-s + 0.0581·26-s + 0.192·27-s + 0.403·28-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 | \( 1 - 13.5T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.20e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 1.96e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.75e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.60e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 2.78e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.56e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.51e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 4.30e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.37e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.61e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.38e9T + 5.50e17T^{2} \) |
| 43 | \( 1 - 4.91e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 9.26e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.11e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 5.97e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.02e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.24e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.72e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.62e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.00e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 6.06e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.53e11T + 7.15e21T^{2} \) |
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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
−9.929691361184226346293728794126, −9.094653044810240094045043001120, −8.047112556020453492285290991186, −7.47781253085466476324683260981, −5.91227162842972485422160720642, −4.44927058526158075481911258407, −3.80275281049955630981404692488, −3.05603240694597786332462198151, −1.02210316695003791867412328194, 0,
1.02210316695003791867412328194, 3.05603240694597786332462198151, 3.80275281049955630981404692488, 4.44927058526158075481911258407, 5.91227162842972485422160720642, 7.47781253085466476324683260981, 8.047112556020453492285290991186, 9.094653044810240094045043001120, 9.929691361184226346293728794126
