| L(s) = 1 | − 5.14·2-s − 243·3-s − 2.02e3·4-s + 3.78e3·5-s + 1.24e3·6-s − 3.63e4·7-s + 2.09e4·8-s + 5.90e4·9-s − 1.94e4·10-s − 6.71e5·11-s + 4.91e5·12-s + 1.37e6·13-s + 1.86e5·14-s − 9.20e5·15-s + 4.03e6·16-s − 4.33e6·17-s − 3.03e5·18-s + 8.25e6·19-s − 7.66e6·20-s + 8.82e6·21-s + 3.45e6·22-s − 6.13e7·23-s − 5.08e6·24-s − 3.44e7·25-s − 7.07e6·26-s − 1.43e7·27-s + 7.33e7·28-s + ⋯ |
| L(s) = 1 | − 0.113·2-s − 0.577·3-s − 0.987·4-s + 0.542·5-s + 0.0655·6-s − 0.816·7-s + 0.225·8-s + 0.333·9-s − 0.0616·10-s − 1.25·11-s + 0.569·12-s + 1.02·13-s + 0.0927·14-s − 0.313·15-s + 0.961·16-s − 0.739·17-s − 0.0378·18-s + 0.764·19-s − 0.535·20-s + 0.471·21-s + 0.142·22-s − 1.98·23-s − 0.130·24-s − 0.705·25-s − 0.116·26-s − 0.192·27-s + 0.805·28-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
| \(L(6)\) |
\(\approx\) |
\(0.4202515602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4202515602\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 5.14T + 2.04e3T^{2} \) |
| 5 | \( 1 - 3.78e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 3.63e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 6.71e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.37e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 4.33e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 8.25e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 6.13e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 7.51e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 2.73e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 5.71e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 9.81e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 1.36e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.13e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.65e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 4.73e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 1.18e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.10e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 4.85e9T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.22e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 3.61e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 1.60e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.28e11T + 7.15e21T^{2} \) |
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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
−10.25966151743458995529923734467, −9.941325518146693684848527204711, −8.742963418107070680522400786141, −7.74365774326251734420338304966, −6.24670252913444613176073802405, −5.58197227660361192908641745350, −4.42987159640363063958460679389, −3.26364333921940516633246114534, −1.71919393330673666241683597197, −0.30705275236426650246694609403,
0.30705275236426650246694609403, 1.71919393330673666241683597197, 3.26364333921940516633246114534, 4.42987159640363063958460679389, 5.58197227660361192908641745350, 6.24670252913444613176073802405, 7.74365774326251734420338304966, 8.742963418107070680522400786141, 9.941325518146693684848527204711, 10.25966151743458995529923734467
