| L(s) = 1 | + 1.14e4·5-s − 1.68e4·7-s + 5.21e5·11-s − 2.16e5·13-s − 4.69e6·17-s − 1.95e7·19-s + 3.37e7·23-s + 8.25e7·25-s − 3.46e7·29-s + 2.80e8·31-s − 1.92e8·35-s + 3.26e8·37-s − 7.09e8·41-s − 7.33e8·43-s + 7.72e8·47-s + 2.82e8·49-s + 3.45e9·53-s + 5.97e9·55-s + 4.54e9·59-s + 5.97e9·61-s − 2.47e9·65-s + 1.20e10·67-s − 2.88e10·71-s + 1.84e10·73-s − 8.76e9·77-s + 4.27e10·79-s + 3.54e9·83-s + ⋯ |
| L(s) = 1 | + 1.64·5-s − 0.377·7-s + 0.976·11-s − 0.161·13-s − 0.802·17-s − 1.81·19-s + 1.09·23-s + 1.69·25-s − 0.313·29-s + 1.75·31-s − 0.619·35-s + 0.774·37-s − 0.955·41-s − 0.760·43-s + 0.491·47-s + 0.142·49-s + 1.13·53-s + 1.60·55-s + 0.827·59-s + 0.905·61-s − 0.265·65-s + 1.09·67-s − 1.89·71-s + 1.04·73-s − 0.369·77-s + 1.56·79-s + 0.0986·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.326616665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.326616665\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 1.68e4T \) |
| good | 5 | \( 1 - 1.14e4T + 4.88e7T^{2} \) |
| 11 | \( 1 - 5.21e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.16e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 4.69e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.95e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.37e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 3.46e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.80e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.26e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 7.09e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 7.33e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 7.72e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.45e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.54e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 5.97e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.20e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.88e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.84e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.27e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.54e9T + 1.28e21T^{2} \) |
| 89 | \( 1 - 7.61e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 8.60e10T + 7.15e21T^{2} \) |
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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
−10.02982382813494199111239948439, −9.215673860329279830872456697066, −8.505644409316452313345717091244, −6.63182495788714849782754017445, −6.47832196171033168487225466743, −5.23657109695690200684098263450, −4.13310578289634774154632433359, −2.65755785057269171861967807777, −1.89279802150897419840604612067, −0.77488201836757995558787359170,
0.77488201836757995558787359170, 1.89279802150897419840604612067, 2.65755785057269171861967807777, 4.13310578289634774154632433359, 5.23657109695690200684098263450, 6.47832196171033168487225466743, 6.63182495788714849782754017445, 8.505644409316452313345717091244, 9.215673860329279830872456697066, 10.02982382813494199111239948439
