| L(s) = 1 | − 6.41·2-s − 81·3-s − 470.·4-s + 625·5-s + 519.·6-s − 7.66e3·7-s + 6.30e3·8-s + 6.56e3·9-s − 4.00e3·10-s + 3.71e3·11-s + 3.81e4·12-s + 1.40e5·13-s + 4.91e4·14-s − 5.06e4·15-s + 2.00e5·16-s − 2.94e4·17-s − 4.20e4·18-s − 1.30e5·19-s − 2.94e5·20-s + 6.20e5·21-s − 2.38e4·22-s + 5.43e5·23-s − 5.10e5·24-s + 3.90e5·25-s − 8.97e5·26-s − 5.31e5·27-s + 3.60e6·28-s + ⋯ |
| L(s) = 1 | − 0.283·2-s − 0.577·3-s − 0.919·4-s + 0.447·5-s + 0.163·6-s − 1.20·7-s + 0.544·8-s + 0.333·9-s − 0.126·10-s + 0.0765·11-s + 0.530·12-s + 1.35·13-s + 0.341·14-s − 0.258·15-s + 0.765·16-s − 0.0854·17-s − 0.0944·18-s − 0.229·19-s − 0.411·20-s + 0.696·21-s − 0.0216·22-s + 0.404·23-s − 0.314·24-s + 0.200·25-s − 0.385·26-s − 0.192·27-s + 1.10·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.8294947967\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8294947967\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 19 | \( 1 + 1.30e5T \) |
| good | 2 | \( 1 + 6.41T + 512T^{2} \) |
| 7 | \( 1 + 7.66e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.71e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.40e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.94e4T + 1.18e11T^{2} \) |
| 23 | \( 1 - 5.43e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 6.53e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 6.20e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.43e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.40e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.38e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 4.14e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.45e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 7.94e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 3.29e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.96e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.62e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.73e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.95e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.00e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.00e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.03e9T + 7.60e17T^{2} \) |
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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
−10.24420256842622118014229201573, −9.242270727346554066318303738907, −8.705127510610751171542761257888, −7.26505326812248557010605115188, −6.21012025284651789196901280147, −5.47762940496209556894276104519, −4.18947024293864303287246607980, −3.25709185327697896925250750672, −1.52101642616700767367612613966, −0.46497635303538942817085906856,
0.46497635303538942817085906856, 1.52101642616700767367612613966, 3.25709185327697896925250750672, 4.18947024293864303287246607980, 5.47762940496209556894276104519, 6.21012025284651789196901280147, 7.26505326812248557010605115188, 8.705127510610751171542761257888, 9.242270727346554066318303738907, 10.24420256842622118014229201573
