| L(s) = 1 | − 2.64·2-s + 5.00·4-s − 5-s − 3.64·7-s − 7.93·8-s + 2.64·10-s − 5.64·11-s − 5.64·13-s + 9.64·14-s + 11.0·16-s + 4·17-s − 19-s − 5.00·20-s + 14.9·22-s + 1.29·23-s + 25-s + 14.9·26-s − 18.2·28-s + 6.93·29-s + 6·31-s − 13.2·32-s − 10.5·34-s + 3.64·35-s − 1.64·37-s + 2.64·38-s + 7.93·40-s + 4.35·41-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 2.50·4-s − 0.447·5-s − 1.37·7-s − 2.80·8-s + 0.836·10-s − 1.70·11-s − 1.56·13-s + 2.57·14-s + 2.75·16-s + 0.970·17-s − 0.229·19-s − 1.11·20-s + 3.18·22-s + 0.269·23-s + 0.200·25-s + 2.92·26-s − 3.44·28-s + 1.28·29-s + 1.07·31-s − 2.33·32-s − 1.81·34-s + 0.616·35-s − 0.270·37-s + 0.429·38-s + 1.25·40-s + 0.680·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2688621685\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2688621685\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 7 | \( 1 + 3.64T + 7T^{2} \) |
| 11 | \( 1 + 5.64T + 11T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 - 6.93T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 + 1.64T + 37T^{2} \) |
| 41 | \( 1 - 4.35T + 41T^{2} \) |
| 43 | \( 1 - 0.354T + 43T^{2} \) |
| 47 | \( 1 + 9.29T + 47T^{2} \) |
| 53 | \( 1 + 0.708T + 53T^{2} \) |
| 59 | \( 1 + 0.708T + 59T^{2} \) |
| 61 | \( 1 + 0.708T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 1.06T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−10.02581513974420537155598790993, −9.579828267433626812097760759297, −8.419923197513881581540426457735, −7.80062728705847226041248418507, −7.11459669919728983154610827023, −6.29703094192450178260044183745, −5.05133025326034824774734512449, −3.08520710724224944364398989554, −2.50442802574960914334913958952, −0.50982924289492545929628281306,
0.50982924289492545929628281306, 2.50442802574960914334913958952, 3.08520710724224944364398989554, 5.05133025326034824774734512449, 6.29703094192450178260044183745, 7.11459669919728983154610827023, 7.80062728705847226041248418507, 8.419923197513881581540426457735, 9.579828267433626812097760759297, 10.02581513974420537155598790993
