| L(s) = 1 | − 63.7·2-s − 283.·3-s + 2.01e3·4-s + 1.80e4·6-s − 4.19e4·7-s + 2.20e3·8-s − 9.69e4·9-s − 9.57e5·11-s − 5.70e5·12-s − 1.39e6·13-s + 2.67e6·14-s − 4.26e6·16-s − 3.76e6·17-s + 6.17e6·18-s + 9.41e6·19-s + 1.18e7·21-s + 6.10e7·22-s − 3.02e7·23-s − 6.24e5·24-s + 8.86e7·26-s + 7.76e7·27-s − 8.44e7·28-s + 1.03e8·29-s − 5.48e7·31-s + 2.67e8·32-s + 2.71e8·33-s + 2.40e8·34-s + ⋯ |
| L(s) = 1 | − 1.40·2-s − 0.672·3-s + 0.983·4-s + 0.947·6-s − 0.942·7-s + 0.0237·8-s − 0.547·9-s − 1.79·11-s − 0.661·12-s − 1.03·13-s + 1.32·14-s − 1.01·16-s − 0.643·17-s + 0.770·18-s + 0.871·19-s + 0.634·21-s + 2.52·22-s − 0.981·23-s − 0.0160·24-s + 1.46·26-s + 1.04·27-s − 0.926·28-s + 0.937·29-s − 0.344·31-s + 1.40·32-s + 1.20·33-s + 0.906·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(0.1274150988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1274150988\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 63.7T + 2.04e3T^{2} \) |
| 3 | \( 1 + 283.T + 1.77e5T^{2} \) |
| 7 | \( 1 + 4.19e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 9.57e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.39e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 3.76e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 9.41e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.02e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.03e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 5.48e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.78e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 9.29e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 2.68e7T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.20e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.02e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 7.97e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 2.07e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 5.61e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.51e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 6.64e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 1.57e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 2.04e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.21e10T + 2.77e21T^{2} \) |
| 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
−15.68664509440942668716342900458, −13.58358617534176262733253986652, −12.09103429721113413941776273738, −10.62440052727329407562524466579, −9.829110845704631110431037691593, −8.329694288214976881922407059345, −6.99335364249493061301882880319, −5.27816277417329085781471990121, −2.55209493283170117722763473654, −0.29362508716115768650175800344,
0.29362508716115768650175800344, 2.55209493283170117722763473654, 5.27816277417329085781471990121, 6.99335364249493061301882880319, 8.329694288214976881922407059345, 9.829110845704631110431037691593, 10.62440052727329407562524466579, 12.09103429721113413941776273738, 13.58358617534176262733253986652, 15.68664509440942668716342900458
