| L(s) = 1 | + 243·3-s − 9.84e3·5-s + 2.95e4·7-s + 5.90e4·9-s + 9.86e5·11-s − 1.64e6·13-s − 2.39e6·15-s + 6.62e6·17-s + 1.17e7·19-s + 7.17e6·21-s + 2.55e7·23-s + 4.80e7·25-s + 1.43e7·27-s + 7.42e7·29-s + 1.43e8·31-s + 2.39e8·33-s − 2.90e8·35-s − 2.05e8·37-s − 3.99e8·39-s + 1.27e9·41-s − 9.67e8·43-s − 5.81e8·45-s − 1.21e9·47-s − 1.10e9·49-s + 1.60e9·51-s − 2.91e9·53-s − 9.70e9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.40·5-s + 0.663·7-s + 0.333·9-s + 1.84·11-s − 1.22·13-s − 0.813·15-s + 1.13·17-s + 1.09·19-s + 0.383·21-s + 0.826·23-s + 0.983·25-s + 0.192·27-s + 0.672·29-s + 0.898·31-s + 1.06·33-s − 0.934·35-s − 0.487·37-s − 0.709·39-s + 1.72·41-s − 1.00·43-s − 0.469·45-s − 0.775·47-s − 0.559·49-s + 0.653·51-s − 0.958·53-s − 2.60·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(2.123642391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.123642391\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 243T \) |
| good | 5 | \( 1 + 9.84e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 2.95e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 9.86e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.64e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 6.62e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.17e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.55e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 7.42e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.43e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.05e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.27e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 9.67e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.21e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.91e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 3.21e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 1.08e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 5.54e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 6.46e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 2.95e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 8.63e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 4.56e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.90e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 9.24e10T + 7.15e21T^{2} \) |
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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
−14.86135714242365586246319658536, −14.28699980980376755780138736393, −12.21148791646861823027062280298, −11.54035117074008037758724370725, −9.588043593241444440730094408793, −8.159472730921378742952300785328, −7.12177568608404639036523176940, −4.62836379891546441358457336381, −3.31432770817561863990762792167, −1.09100369337442507930503384277,
1.09100369337442507930503384277, 3.31432770817561863990762792167, 4.62836379891546441358457336381, 7.12177568608404639036523176940, 8.159472730921378742952300785328, 9.588043593241444440730094408793, 11.54035117074008037758724370725, 12.21148791646861823027062280298, 14.28699980980376755780138736393, 14.86135714242365586246319658536
