| L(s) = 1 | + 1.05e4·5-s + 3.64e4·7-s − 2.86e5·11-s − 1.97e6·13-s + 1.34e6·17-s + 1.68e7·19-s + 3.95e7·23-s + 6.27e7·25-s + 1.70e8·29-s − 1.59e8·31-s + 3.85e8·35-s − 2.92e8·37-s − 2.22e8·41-s + 8.28e8·43-s + 6.71e8·47-s − 6.48e8·49-s − 3.76e9·53-s − 3.03e9·55-s − 1.13e9·59-s + 9.13e9·61-s − 2.08e10·65-s + 8.39e9·67-s + 2.73e10·71-s − 1.62e10·73-s − 1.04e10·77-s + 1.95e10·79-s + 4.79e10·83-s + ⋯ |
| L(s) = 1 | + 1.51·5-s + 0.819·7-s − 0.537·11-s − 1.47·13-s + 0.229·17-s + 1.56·19-s + 1.28·23-s + 1.28·25-s + 1.54·29-s − 1.00·31-s + 1.23·35-s − 0.693·37-s − 0.300·41-s + 0.859·43-s + 0.427·47-s − 0.327·49-s − 1.23·53-s − 0.811·55-s − 0.207·59-s + 1.38·61-s − 2.22·65-s + 0.759·67-s + 1.79·71-s − 0.915·73-s − 0.440·77-s + 0.716·79-s + 1.33·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(3.457430108\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.457430108\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.05e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.64e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 2.86e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.97e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.34e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.68e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 3.95e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 1.70e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.59e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.92e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 2.22e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 8.28e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 6.71e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.76e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 1.13e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 9.13e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 8.39e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.73e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.62e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.95e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.79e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 2.86e9T + 2.77e21T^{2} \) |
| 97 | \( 1 - 6.47e10T + 7.15e21T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.84709380781366376862700380918, −9.913729654100494335298439170469, −9.205309321916639519767826333106, −7.83448411675412657729886145821, −6.81456276176915912878331360185, −5.33408253897021989392462519551, −5.00518225661839339093114826712, −2.94789604103584933305580191471, −2.01143900556977213023236143024, −0.912909153250063435981704558501,
0.912909153250063435981704558501, 2.01143900556977213023236143024, 2.94789604103584933305580191471, 5.00518225661839339093114826712, 5.33408253897021989392462519551, 6.81456276176915912878331360185, 7.83448411675412657729886145821, 9.205309321916639519767826333106, 9.913729654100494335298439170469, 10.84709380781366376862700380918
