| L(s) = 1 | − 88.0·2-s − 243·3-s + 5.70e3·4-s − 1.34e4·5-s + 2.14e4·6-s + 1.68e4·7-s − 3.22e5·8-s + 5.90e4·9-s + 1.18e6·10-s + 9.66e5·11-s − 1.38e6·12-s − 3.71e5·13-s − 1.48e6·14-s + 3.25e6·15-s + 1.67e7·16-s − 3.26e6·17-s − 5.20e6·18-s + 7.04e6·19-s − 7.65e7·20-s − 4.08e6·21-s − 8.51e7·22-s − 2.58e7·23-s + 7.83e7·24-s + 1.30e8·25-s + 3.27e7·26-s − 1.43e7·27-s + 9.59e7·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 0.577·3-s + 2.78·4-s − 1.91·5-s + 1.12·6-s + 0.377·7-s − 3.47·8-s + 0.333·9-s + 3.73·10-s + 1.80·11-s − 1.60·12-s − 0.277·13-s − 0.735·14-s + 1.10·15-s + 3.98·16-s − 0.558·17-s − 0.648·18-s + 0.652·19-s − 5.34·20-s − 0.218·21-s − 3.52·22-s − 0.836·23-s + 2.00·24-s + 2.67·25-s + 0.539·26-s − 0.192·27-s + 1.05·28-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 | \( 1 + 88.0T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.34e4T + 4.88e7T^{2} \) |
| 11 | \( 1 - 9.66e5T + 2.85e11T^{2} \) |
| 17 | \( 1 + 3.26e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 7.04e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.58e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 9.40e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.08e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.39e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 6.21e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.48e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.09e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.34e8T + 9.26e18T^{2} \) |
| 59 | \( 1 + 7.67e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 1.30e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.72e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.59e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 9.41e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.58e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.74e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 3.74e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 3.78e10T + 7.15e21T^{2} \) |
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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
−9.287019433234962638610899969180, −8.626121023746689997532183368920, −7.62988128121361934575460360130, −7.13861180642953656766950600691, −6.17716649551105149785976090824, −4.34397542989179058898335040290, −3.29245424626590567049222695598, −1.67904639975996568713849948881, −0.795376565750484546912950739255, 0,
0.795376565750484546912950739255, 1.67904639975996568713849948881, 3.29245424626590567049222695598, 4.34397542989179058898335040290, 6.17716649551105149785976090824, 7.13861180642953656766950600691, 7.62988128121361934575460360130, 8.626121023746689997532183368920, 9.287019433234962638610899969180
