| L(s) = 1 | + 37.7·2-s + 83.4·3-s + 912.·4-s + 1.39e3·5-s + 3.14e3·6-s + 8.64e3·7-s + 1.51e4·8-s − 1.27e4·9-s + 5.27e4·10-s − 5.11e4·11-s + 7.61e4·12-s − 2.57e4·13-s + 3.26e5·14-s + 1.16e5·15-s + 1.02e5·16-s − 4.79e5·17-s − 4.80e5·18-s + 1.08e6·19-s + 1.27e6·20-s + 7.21e5·21-s − 1.93e6·22-s + 2.01e6·23-s + 1.26e6·24-s + 3.06e3·25-s − 9.71e5·26-s − 2.70e6·27-s + 7.88e6·28-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 0.594·3-s + 1.78·4-s + 1.00·5-s + 0.991·6-s + 1.36·7-s + 1.30·8-s − 0.646·9-s + 1.66·10-s − 1.05·11-s + 1.05·12-s − 0.249·13-s + 2.26·14-s + 0.595·15-s + 0.392·16-s − 1.39·17-s − 1.07·18-s + 1.91·19-s + 1.78·20-s + 0.809·21-s − 1.75·22-s + 1.50·23-s + 0.775·24-s + 0.00157·25-s − 0.416·26-s − 0.979·27-s + 2.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(7.010212063\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.010212063\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 3.41e6T \) |
| good | 2 | \( 1 - 37.7T + 512T^{2} \) |
| 3 | \( 1 - 83.4T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.39e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.64e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.11e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 2.57e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 4.79e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 1.08e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.01e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.78e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.50e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.04e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.11e7T + 3.27e14T^{2} \) |
| 47 | \( 1 - 3.87e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.65e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 5.14e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.84e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 4.03e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 5.56e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.12e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.49e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.20e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.34e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 2.94e7T + 7.60e17T^{2} \) |
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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
−13.74670433581476966729982878501, −13.43949522261972008226973969190, −11.82475610507456679031809015710, −10.81351163777251731541094526666, −8.945552699969861506650693585294, −7.35581858978774778083191826035, −5.58067875612130082961477153199, −4.89310253404278953330709569866, −2.99005955841630018210330326432, −1.97054421464227049345754694803,
1.97054421464227049345754694803, 2.99005955841630018210330326432, 4.89310253404278953330709569866, 5.58067875612130082961477153199, 7.35581858978774778083191826035, 8.945552699969861506650693585294, 10.81351163777251731541094526666, 11.82475610507456679031809015710, 13.43949522261972008226973969190, 13.74670433581476966729982878501
