| L(s) = 1 | − 41.9·2-s − 243·3-s − 287.·4-s − 4.79e3·5-s + 1.01e4·6-s − 4.09e4·7-s + 9.79e4·8-s + 5.90e4·9-s + 2.01e5·10-s + 2.61e5·11-s + 6.99e4·12-s − 3.20e5·13-s + 1.71e6·14-s + 1.16e6·15-s − 3.52e6·16-s + 8.00e6·17-s − 2.47e6·18-s + 4.69e6·19-s + 1.38e6·20-s + 9.95e6·21-s − 1.09e7·22-s + 2.01e7·23-s − 2.38e7·24-s − 2.57e7·25-s + 1.34e7·26-s − 1.43e7·27-s + 1.17e7·28-s + ⋯ |
| L(s) = 1 | − 0.927·2-s − 0.577·3-s − 0.140·4-s − 0.686·5-s + 0.535·6-s − 0.921·7-s + 1.05·8-s + 0.333·9-s + 0.636·10-s + 0.490·11-s + 0.0811·12-s − 0.239·13-s + 0.854·14-s + 0.396·15-s − 0.839·16-s + 1.36·17-s − 0.309·18-s + 0.434·19-s + 0.0964·20-s + 0.531·21-s − 0.454·22-s + 0.651·23-s − 0.610·24-s − 0.528·25-s + 0.221·26-s − 0.192·27-s + 0.129·28-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
| \(L(6)\) |
\(\approx\) |
\(0.5286463992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5286463992\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 41.9T + 2.04e3T^{2} \) |
| 5 | \( 1 + 4.79e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 4.09e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.61e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 3.20e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 8.00e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 4.69e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.01e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 9.30e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 4.66e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.29e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.99e7T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.68e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 5.76e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.39e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 2.27e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.03e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.02e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 3.07e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 2.51e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.75e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.52e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.56e10T + 7.15e21T^{2} \) |
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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
−10.33938660547965579499276278243, −9.772181634886022371169034596445, −8.749289977306908856320004007183, −7.66867371357843276385091320350, −6.85446407209584277728280889603, −5.52414294919680455447192983871, −4.29429258232598763568506148496, −3.20550055115224699667410019946, −1.33291719416143670022976926804, −0.44304840143787358746357002310,
0.44304840143787358746357002310, 1.33291719416143670022976926804, 3.20550055115224699667410019946, 4.29429258232598763568506148496, 5.52414294919680455447192983871, 6.85446407209584277728280889603, 7.66867371357843276385091320350, 8.749289977306908856320004007183, 9.772181634886022371169034596445, 10.33938660547965579499276278243
