| L(s) = 1 | + 63.4·2-s − 243·3-s + 1.98e3·4-s + 1.29e4·5-s − 1.54e4·6-s + 1.68e4·7-s − 4.29e3·8-s + 5.90e4·9-s + 8.21e5·10-s − 3.34e5·11-s − 4.81e5·12-s − 3.71e5·13-s + 1.06e6·14-s − 3.14e6·15-s − 4.32e6·16-s + 3.98e6·17-s + 3.74e6·18-s − 1.30e7·19-s + 2.56e7·20-s − 4.08e6·21-s − 2.12e7·22-s − 1.67e7·23-s + 1.04e6·24-s + 1.18e8·25-s − 2.35e7·26-s − 1.43e7·27-s + 3.32e7·28-s + ⋯ |
| L(s) = 1 | + 1.40·2-s − 0.577·3-s + 0.966·4-s + 1.85·5-s − 0.809·6-s + 0.377·7-s − 0.0463·8-s + 0.333·9-s + 2.59·10-s − 0.625·11-s − 0.558·12-s − 0.277·13-s + 0.530·14-s − 1.06·15-s − 1.03·16-s + 0.680·17-s + 0.467·18-s − 1.20·19-s + 1.79·20-s − 0.218·21-s − 0.877·22-s − 0.543·23-s + 0.0267·24-s + 2.43·25-s − 0.388·26-s − 0.192·27-s + 0.365·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 243T \) |
| 7 | \( 1 - 1.68e4T \) |
| 13 | \( 1 + 3.71e5T \) |
| good | 2 | \( 1 - 63.4T + 2.04e3T^{2} \) |
| 5 | \( 1 - 1.29e4T + 4.88e7T^{2} \) |
| 11 | \( 1 + 3.34e5T + 2.85e11T^{2} \) |
| 17 | \( 1 - 3.98e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.30e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.67e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.03e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 2.10e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.90e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 2.66e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.44e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.10e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.70e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 2.51e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.07e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.80e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.98e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.68e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.07e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 4.80e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.82e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 2.89e10T + 7.15e21T^{2} \) |
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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
−9.808912583572374669754668257343, −8.667254132183324793119807314960, −7.03303721724746377535241748175, −6.13676265384980048012889951213, −5.41046227980022382531615200478, −4.97596333066837824904285228119, −3.61796805536683410400426226413, −2.30917751213783453644774870802, −1.69704781679607037733977066805, 0,
1.69704781679607037733977066805, 2.30917751213783453644774870802, 3.61796805536683410400426226413, 4.97596333066837824904285228119, 5.41046227980022382531615200478, 6.13676265384980048012889951213, 7.03303721724746377535241748175, 8.667254132183324793119807314960, 9.808912583572374669754668257343
