| L(s) = 1 | − 70.9·2-s − 243·3-s + 2.98e3·4-s + 3.03e3·5-s + 1.72e4·6-s − 6.20e4·7-s − 6.67e4·8-s + 5.90e4·9-s − 2.15e5·10-s + 2.99e5·11-s − 7.26e5·12-s − 1.56e6·13-s + 4.40e6·14-s − 7.36e5·15-s − 1.38e6·16-s − 5.69e5·17-s − 4.19e6·18-s − 2.09e7·19-s + 9.06e6·20-s + 1.50e7·21-s − 2.12e7·22-s + 2.52e7·23-s + 1.62e7·24-s − 3.96e7·25-s + 1.10e8·26-s − 1.43e7·27-s − 1.85e8·28-s + ⋯ |
| L(s) = 1 | − 1.56·2-s − 0.577·3-s + 1.45·4-s + 0.433·5-s + 0.905·6-s − 1.39·7-s − 0.719·8-s + 0.333·9-s − 0.680·10-s + 0.560·11-s − 0.842·12-s − 1.16·13-s + 2.18·14-s − 0.250·15-s − 0.330·16-s − 0.0972·17-s − 0.522·18-s − 1.93·19-s + 0.633·20-s + 0.805·21-s − 0.878·22-s + 0.816·23-s + 0.415·24-s − 0.811·25-s + 1.83·26-s − 0.192·27-s − 2.03·28-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
| \(L(6)\) |
\(\approx\) |
\(0.06598451389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06598451389\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 70.9T + 2.04e3T^{2} \) |
| 5 | \( 1 - 3.03e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.20e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 2.99e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 1.56e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 5.69e5T + 3.42e13T^{2} \) |
| 19 | \( 1 + 2.09e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 2.52e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 8.20e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.20e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.75e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 3.35e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.05e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 5.28e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 5.65e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 7.64e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 9.27e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 1.76e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 8.95e8T + 3.13e20T^{2} \) |
| 79 | \( 1 - 1.22e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 6.96e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 1.97e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 1.55e11T + 7.15e21T^{2} \) |
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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
−10.39214747234498225734688684648, −9.529710953172202825167898312236, −9.100795325891146137156163350708, −7.63393081424028073159492078870, −6.71655135890073388623539583264, −5.99968281887459369458754599822, −4.31673720439375572974514812517, −2.64440035966447355871739673200, −1.57084116695392349369424803890, −0.15099002271638858402024158444,
0.15099002271638858402024158444, 1.57084116695392349369424803890, 2.64440035966447355871739673200, 4.31673720439375572974514812517, 5.99968281887459369458754599822, 6.71655135890073388623539583264, 7.63393081424028073159492078870, 9.100795325891146137156163350708, 9.529710953172202825167898312236, 10.39214747234498225734688684648
