L(s) = 1 | + 16.4·2-s − 243·3-s − 1.77e3·4-s − 1.11e4·5-s − 3.99e3·6-s − 6.47e4·7-s − 6.29e4·8-s + 5.90e4·9-s − 1.83e5·10-s − 1.52e5·11-s + 4.31e5·12-s + 2.10e6·13-s − 1.06e6·14-s + 2.71e6·15-s + 2.60e6·16-s − 1.11e7·17-s + 9.71e5·18-s − 1.27e7·19-s + 1.98e7·20-s + 1.57e7·21-s − 2.51e6·22-s + 9.67e6·23-s + 1.52e7·24-s + 7.59e7·25-s + 3.47e7·26-s − 1.43e7·27-s + 1.15e8·28-s + ⋯ |
L(s) = 1 | + 0.363·2-s − 0.577·3-s − 0.867·4-s − 1.59·5-s − 0.209·6-s − 1.45·7-s − 0.679·8-s + 0.333·9-s − 0.581·10-s − 0.286·11-s + 0.500·12-s + 1.57·13-s − 0.529·14-s + 0.922·15-s + 0.620·16-s − 1.90·17-s + 0.121·18-s − 1.18·19-s + 1.38·20-s + 0.840·21-s − 0.104·22-s + 0.313·23-s + 0.392·24-s + 1.55·25-s + 0.573·26-s − 0.192·27-s + 1.26·28-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 | \( 1 - 16.4T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.11e4T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.47e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 1.52e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.10e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 1.11e7T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.27e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 9.67e6T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.78e8T + 1.22e16T^{2} \) |
| 31 | \( 1 - 8.93e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.78e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.14e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 8.43e7T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.29e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.97e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 6.12e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.63e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.19e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.52e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.83e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.79e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 4.73e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.10e11T + 7.15e21T^{2} \) |
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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
−10.37728815888642686400883858124, −8.922989648689079075116081755207, −8.372381206599976001427373150446, −6.80046812611224523408503732140, −6.12799709578067003027560915015, −4.52519970997428366096835051871, −3.99568306176289067551321785481, −3.00153850347900188835458722677, −0.68399124128263829123891802870, 0,
0.68399124128263829123891802870, 3.00153850347900188835458722677, 3.99568306176289067551321785481, 4.52519970997428366096835051871, 6.12799709578067003027560915015, 6.80046812611224523408503732140, 8.372381206599976001427373150446, 8.922989648689079075116081755207, 10.37728815888642686400883858124