| L(s) = 1 | + 65.6·2-s + 243·3-s + 2.26e3·4-s − 3.29e3·5-s + 1.59e4·6-s + 5.47e4·7-s + 1.44e4·8-s + 5.90e4·9-s − 2.16e5·10-s − 4.23e5·11-s + 5.50e5·12-s − 1.58e6·13-s + 3.59e6·14-s − 8.00e5·15-s − 3.69e6·16-s − 4.57e6·17-s + 3.87e6·18-s + 1.41e7·19-s − 7.46e6·20-s + 1.32e7·21-s − 2.78e7·22-s − 1.71e7·23-s + 3.50e6·24-s − 3.79e7·25-s − 1.04e8·26-s + 1.43e7·27-s + 1.24e8·28-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 0.577·3-s + 1.10·4-s − 0.471·5-s + 0.838·6-s + 1.23·7-s + 0.155·8-s + 0.333·9-s − 0.684·10-s − 0.793·11-s + 0.639·12-s − 1.18·13-s + 1.78·14-s − 0.272·15-s − 0.881·16-s − 0.781·17-s + 0.483·18-s + 1.31·19-s − 0.521·20-s + 0.710·21-s − 1.15·22-s − 0.554·23-s + 0.0897·24-s − 0.777·25-s − 1.71·26-s + 0.192·27-s + 1.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 243T \) |
| 59 | \( 1 + 7.14e8T \) |
| good | 2 | \( 1 - 65.6T + 2.04e3T^{2} \) |
| 5 | \( 1 + 3.29e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 5.47e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 4.23e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.58e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 4.57e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.41e7T + 1.16e14T^{2} \) |
| 23 | \( 1 + 1.71e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 5.00e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.79e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.12e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.65e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 8.33e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.33e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 8.94e8T + 9.26e18T^{2} \) |
| 61 | \( 1 - 7.14e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.42e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.36e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.92e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 2.75e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.14e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.56e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.05e11T + 7.15e21T^{2} \) |
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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
−10.41561744922112426129163896265, −9.041479936534016045752534989897, −7.85170027932225502202008566280, −7.15689600857885999219129180829, −5.50442573819868559810160672459, −4.81927463630784014169559919563, −3.91446493101410325044712308277, −2.76383506646131365276678382274, −1.85574121300937972195570664559, 0,
1.85574121300937972195570664559, 2.76383506646131365276678382274, 3.91446493101410325044712308277, 4.81927463630784014169559919563, 5.50442573819868559810160672459, 7.15689600857885999219129180829, 7.85170027932225502202008566280, 9.041479936534016045752534989897, 10.41561744922112426129163896265
