| L(s) = 1 | + 11·2-s + 9·3-s + 89·4-s + 6·5-s + 99·6-s − 176·7-s + 627·8-s + 81·9-s + 66·10-s − 496·11-s + 801·12-s − 178·13-s − 1.93e3·14-s + 54·15-s + 4.04e3·16-s + 202·17-s + 891·18-s − 361·19-s + 534·20-s − 1.58e3·21-s − 5.45e3·22-s + 4.39e3·23-s + 5.64e3·24-s − 3.08e3·25-s − 1.95e3·26-s + 729·27-s − 1.56e4·28-s + ⋯ |
| L(s) = 1 | + 1.94·2-s + 0.577·3-s + 2.78·4-s + 0.107·5-s + 1.12·6-s − 1.35·7-s + 3.46·8-s + 1/3·9-s + 0.208·10-s − 1.23·11-s + 1.60·12-s − 0.292·13-s − 2.63·14-s + 0.0619·15-s + 3.95·16-s + 0.169·17-s + 0.648·18-s − 0.229·19-s + 0.298·20-s − 0.783·21-s − 2.40·22-s + 1.73·23-s + 1.99·24-s − 0.988·25-s − 0.568·26-s + 0.192·27-s − 3.77·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.511210796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.511210796\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{2} T \) |
| 19 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 - 11 T + p^{5} T^{2} \) |
| 5 | \( 1 - 6 T + p^{5} T^{2} \) |
| 7 | \( 1 + 176 T + p^{5} T^{2} \) |
| 11 | \( 1 + 496 T + p^{5} T^{2} \) |
| 13 | \( 1 + 178 T + p^{5} T^{2} \) |
| 17 | \( 1 - 202 T + p^{5} T^{2} \) |
| 23 | \( 1 - 4396 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5902 T + p^{5} T^{2} \) |
| 31 | \( 1 - 5760 T + p^{5} T^{2} \) |
| 37 | \( 1 + 3906 T + p^{5} T^{2} \) |
| 41 | \( 1 - 15774 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7492 T + p^{5} T^{2} \) |
| 47 | \( 1 + 7452 T + p^{5} T^{2} \) |
| 53 | \( 1 + 29014 T + p^{5} T^{2} \) |
| 59 | \( 1 - 13604 T + p^{5} T^{2} \) |
| 61 | \( 1 + 12466 T + p^{5} T^{2} \) |
| 67 | \( 1 - 43436 T + p^{5} T^{2} \) |
| 71 | \( 1 - 28800 T + p^{5} T^{2} \) |
| 73 | \( 1 - 80746 T + p^{5} T^{2} \) |
| 79 | \( 1 - 76456 T + p^{5} T^{2} \) |
| 83 | \( 1 + 56880 T + p^{5} T^{2} \) |
| 89 | \( 1 + 103266 T + p^{5} T^{2} \) |
| 97 | \( 1 - 82490 T + p^{5} T^{2} \) |
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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
−13.98800397474863608599675791227, −13.03722242831965531774984139693, −12.67616411506978472665099460367, −11.09123673955868331297614266307, −9.832383616215467997817464107279, −7.60618535240269433099969921689, −6.43027415481309161420455823857, −5.11860531779210809528511092149, −3.51796752996316628780149244459, −2.51767540315829687606114375759,
2.51767540315829687606114375759, 3.51796752996316628780149244459, 5.11860531779210809528511092149, 6.43027415481309161420455823857, 7.60618535240269433099969921689, 9.832383616215467997817464107279, 11.09123673955868331297614266307, 12.67616411506978472665099460367, 13.03722242831965531774984139693, 13.98800397474863608599675791227
