| L(s) = 1 | − 33.4·2-s + 243·3-s − 927.·4-s − 1.18e4·5-s − 8.13e3·6-s + 5.53e4·7-s + 9.96e4·8-s + 5.90e4·9-s + 3.95e5·10-s + 3.10e5·11-s − 2.25e5·12-s + 6.35e5·13-s − 1.85e6·14-s − 2.87e6·15-s − 1.43e6·16-s + 4.94e6·17-s − 1.97e6·18-s + 7.38e6·19-s + 1.09e7·20-s + 1.34e7·21-s − 1.03e7·22-s + 5.49e7·23-s + 2.42e7·24-s + 9.10e7·25-s − 2.12e7·26-s + 1.43e7·27-s − 5.12e7·28-s + ⋯ |
| L(s) = 1 | − 0.739·2-s + 0.577·3-s − 0.452·4-s − 1.69·5-s − 0.427·6-s + 1.24·7-s + 1.07·8-s + 0.333·9-s + 1.25·10-s + 0.580·11-s − 0.261·12-s + 0.474·13-s − 0.919·14-s − 0.977·15-s − 0.341·16-s + 0.845·17-s − 0.246·18-s + 0.684·19-s + 0.766·20-s + 0.718·21-s − 0.429·22-s + 1.78·23-s + 0.620·24-s + 1.86·25-s − 0.350·26-s + 0.192·27-s − 0.563·28-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
| \(L(6)\) |
\(\approx\) |
\(1.593941299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593941299\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 + 33.4T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.18e4T + 4.88e7T^{2} \) |
| 7 | \( 1 - 5.53e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 3.10e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 6.35e5T + 1.79e12T^{2} \) |
| 17 | \( 1 - 4.94e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 7.38e6T + 1.16e14T^{2} \) |
| 23 | \( 1 - 5.49e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.08e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.74e8T + 2.54e16T^{2} \) |
| 37 | \( 1 - 2.79e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 2.62e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 6.62e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 2.01e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 1.32e9T + 9.26e18T^{2} \) |
| 61 | \( 1 + 1.28e10T + 4.35e19T^{2} \) |
| 67 | \( 1 + 3.98e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 4.51e9T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.35e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.46e10T + 7.47e20T^{2} \) |
| 83 | \( 1 - 3.01e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.22e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 8.94e10T + 7.15e21T^{2} \) |
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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
−10.79101821785775758984229264162, −9.263446186149547931939176457434, −8.635467469814692173709797829916, −7.69418488947505187028808835567, −7.37936955011693033815208270266, −5.12180375982232852596211864983, −4.18201094722301421113370756040, −3.35393173374231250147332843626, −1.45513549032921445203798128304, −0.71262934402515369160505182631,
0.71262934402515369160505182631, 1.45513549032921445203798128304, 3.35393173374231250147332843626, 4.18201094722301421113370756040, 5.12180375982232852596211864983, 7.37936955011693033815208270266, 7.69418488947505187028808835567, 8.635467469814692173709797829916, 9.263446186149547931939176457434, 10.79101821785775758984229264162
