| L(s) = 1 | − 1.64·2-s − 5.37·3-s − 5.31·4-s − 5·5-s + 8.81·6-s + 2.20·7-s + 21.8·8-s + 1.88·9-s + 8.20·10-s + 18.7·11-s + 28.5·12-s + 62.9·13-s − 3.60·14-s + 26.8·15-s + 6.68·16-s − 17·17-s − 3.09·18-s − 47.8·19-s + 26.5·20-s − 11.8·21-s − 30.7·22-s + 153.·23-s − 117.·24-s + 25·25-s − 103.·26-s + 134.·27-s − 11.6·28-s + ⋯ |
| L(s) = 1 | − 0.579·2-s − 1.03·3-s − 0.663·4-s − 0.447·5-s + 0.599·6-s + 0.118·7-s + 0.964·8-s + 0.0698·9-s + 0.259·10-s + 0.514·11-s + 0.686·12-s + 1.34·13-s − 0.0689·14-s + 0.462·15-s + 0.104·16-s − 0.242·17-s − 0.0405·18-s − 0.577·19-s + 0.296·20-s − 0.122·21-s − 0.298·22-s + 1.39·23-s − 0.997·24-s + 0.200·25-s − 0.778·26-s + 0.962·27-s − 0.0788·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5850602726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5850602726\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 17 | \( 1 + 17T \) |
| good | 2 | \( 1 + 1.64T + 8T^{2} \) |
| 3 | \( 1 + 5.37T + 27T^{2} \) |
| 7 | \( 1 - 2.20T + 343T^{2} \) |
| 11 | \( 1 - 18.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.9T + 2.19e3T^{2} \) |
| 19 | \( 1 + 47.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 153.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 64.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 32.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 159.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 111.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 614.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 308.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 267.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 521.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 118.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.14e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 40.7T + 3.89e5T^{2} \) |
| 79 | \( 1 - 374.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 826.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 38.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 917.T + 9.12e5T^{2} \) |
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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
−13.62064953262927678806666962454, −12.57742948317407991377594083788, −11.26615596369562402083065683204, −10.72665792889003534743245662327, −9.179278558063589002750104557368, −8.300035359640679909893669673492, −6.75505135012596944977442189398, −5.37220924724512324166708663936, −4.00385662350735805975507730683, −0.844794642058816184049789383884,
0.844794642058816184049789383884, 4.00385662350735805975507730683, 5.37220924724512324166708663936, 6.75505135012596944977442189398, 8.300035359640679909893669673492, 9.179278558063589002750104557368, 10.72665792889003534743245662327, 11.26615596369562402083065683204, 12.57742948317407991377594083788, 13.62064953262927678806666962454
