| L(s) = 1 | + 65.8·2-s + 243·3-s + 2.28e3·4-s − 9.01e3·5-s + 1.59e4·6-s − 6.05e4·7-s + 1.55e4·8-s + 5.90e4·9-s − 5.93e5·10-s − 2.03e5·11-s + 5.55e5·12-s + 1.02e6·13-s − 3.98e6·14-s − 2.19e6·15-s − 3.65e6·16-s − 7.79e5·17-s + 3.88e6·18-s + 1.22e7·19-s − 2.05e7·20-s − 1.47e7·21-s − 1.33e7·22-s + 1.69e7·23-s + 3.78e6·24-s + 3.24e7·25-s + 6.75e7·26-s + 1.43e7·27-s − 1.38e8·28-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 0.577·3-s + 1.11·4-s − 1.28·5-s + 0.839·6-s − 1.36·7-s + 0.168·8-s + 0.333·9-s − 1.87·10-s − 0.380·11-s + 0.644·12-s + 0.766·13-s − 1.98·14-s − 0.744·15-s − 0.871·16-s − 0.133·17-s + 0.484·18-s + 1.13·19-s − 1.43·20-s − 0.786·21-s − 0.553·22-s + 0.548·23-s + 0.0970·24-s + 0.663·25-s + 1.11·26-s + 0.192·27-s − 1.51·28-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
| \(L(6)\) |
\(\approx\) |
\(3.562274278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.562274278\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 65.8T + 2.04e3T^{2} \) |
| 5 | \( 1 + 9.01e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 6.05e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 2.03e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.02e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 7.79e5T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.22e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 1.69e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 6.11e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.67e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + 6.00e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.81e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 4.15e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + 8.28e7T + 2.47e18T^{2} \) |
| 53 | \( 1 - 2.67e9T + 9.26e18T^{2} \) |
| 61 | \( 1 - 9.04e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 5.14e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 2.17e10T + 2.31e20T^{2} \) |
| 73 | \( 1 - 1.30e10T + 3.13e20T^{2} \) |
| 79 | \( 1 - 9.94e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.98e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 8.04e9T + 2.77e21T^{2} \) |
| 97 | \( 1 + 8.30e10T + 7.15e21T^{2} \) |
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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
−11.03508953808099486155696263456, −9.654036979945436544304936347193, −8.549408369225402243094322859610, −7.34538623443914100539311582012, −6.47126618265060009863047151247, −5.24386975234253853175626712197, −3.97530720984038686514956756498, −3.45483161273430973882112255885, −2.65933731554985311161502536034, −0.64446966702814512877776198841,
0.64446966702814512877776198841, 2.65933731554985311161502536034, 3.45483161273430973882112255885, 3.97530720984038686514956756498, 5.24386975234253853175626712197, 6.47126618265060009863047151247, 7.34538623443914100539311582012, 8.549408369225402243094322859610, 9.654036979945436544304936347193, 11.03508953808099486155696263456
