| L(s) = 1 | + 32·2-s + 745.·3-s + 1.02e3·4-s + 3.12e3·5-s + 2.38e4·6-s − 7.14e4·7-s + 3.27e4·8-s + 3.78e5·9-s + 1.00e5·10-s − 3.45e5·11-s + 7.63e5·12-s + 1.50e6·13-s − 2.28e6·14-s + 2.33e6·15-s + 1.04e6·16-s − 5.39e6·17-s + 1.21e7·18-s − 1.11e7·19-s + 3.20e6·20-s − 5.33e7·21-s − 1.10e7·22-s − 5.27e6·23-s + 2.44e7·24-s + 9.76e6·25-s + 4.83e7·26-s + 1.50e8·27-s − 7.32e7·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.77·3-s + 0.5·4-s + 0.447·5-s + 1.25·6-s − 1.60·7-s + 0.353·8-s + 2.13·9-s + 0.316·10-s − 0.647·11-s + 0.885·12-s + 1.12·13-s − 1.13·14-s + 0.792·15-s + 0.250·16-s − 0.921·17-s + 1.51·18-s − 1.03·19-s + 0.223·20-s − 2.84·21-s − 0.457·22-s − 0.170·23-s + 0.626·24-s + 0.199·25-s + 0.797·26-s + 2.01·27-s − 0.803·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.823244673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.823244673\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 32T \) |
| 5 | \( 1 - 3.12e3T \) |
| good | 3 | \( 1 - 745.T + 1.77e5T^{2} \) |
| 7 | \( 1 + 7.14e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 3.45e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.50e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.39e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + 1.11e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.27e6T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.86e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 7.10e7T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.23e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 9.11e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.16e9T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.81e8T + 2.47e18T^{2} \) |
| 53 | \( 1 - 4.05e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 4.89e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.07e10T + 4.35e19T^{2} \) |
| 67 | \( 1 - 3.70e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 3.45e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.21e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 7.02e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 5.55e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 9.29e9T + 2.77e21T^{2} \) |
| 97 | \( 1 - 4.71e10T + 7.15e21T^{2} \) |
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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
−18.69748957891774686927420549159, −16.08971783516549348062018525335, −15.08580677503117582957814296325, −13.43059715146488194197007047003, −13.11436772873243776789052969339, −10.11655535441403759245036547554, −8.638804591976078545783897909957, −6.60100364447373069342408599248, −3.70379288908772639419701663880, −2.40946196629544146143956038996,
2.40946196629544146143956038996, 3.70379288908772639419701663880, 6.60100364447373069342408599248, 8.638804591976078545783897909957, 10.11655535441403759245036547554, 13.11436772873243776789052969339, 13.43059715146488194197007047003, 15.08580677503117582957814296325, 16.08971783516549348062018525335, 18.69748957891774686927420549159
