| L(s) = 1 | + 28.6·2-s + 308.·4-s − 1.77e3·5-s − 9.59e3·7-s − 5.83e3·8-s − 5.08e4·10-s − 1.86e4·11-s + 1.18e5·13-s − 2.74e5·14-s − 3.24e5·16-s − 8.35e4·17-s + 3.65e5·19-s − 5.47e5·20-s − 5.34e5·22-s − 1.22e6·23-s + 1.20e6·25-s + 3.38e6·26-s − 2.95e6·28-s + 2.55e6·29-s + 8.56e6·31-s − 6.31e6·32-s − 2.39e6·34-s + 1.70e7·35-s − 7.97e6·37-s + 1.04e7·38-s + 1.03e7·40-s + 3.21e7·41-s + ⋯ |
| L(s) = 1 | + 1.26·2-s + 0.602·4-s − 1.27·5-s − 1.51·7-s − 0.503·8-s − 1.60·10-s − 0.384·11-s + 1.14·13-s − 1.91·14-s − 1.23·16-s − 0.242·17-s + 0.643·19-s − 0.765·20-s − 0.486·22-s − 0.914·23-s + 0.616·25-s + 1.45·26-s − 0.909·28-s + 0.670·29-s + 1.66·31-s − 1.06·32-s − 0.306·34-s + 1.92·35-s − 0.699·37-s + 0.814·38-s + 0.640·40-s + 1.77·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.695185294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.695185294\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + 8.35e4T \) |
| good | 2 | \( 1 - 28.6T + 512T^{2} \) |
| 5 | \( 1 + 1.77e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 9.59e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.86e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.18e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 3.65e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.22e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.55e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.56e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 7.97e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 3.21e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.38e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.94e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.96e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.58e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.09e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.19e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.93e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.53e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.36e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.51e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 6.97e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.50e8T + 7.60e17T^{2} \) |
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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
−11.75942563620851620754048103243, −10.48507419193051475897500424088, −9.188235332153868060901437519741, −8.020080629317086452733830379356, −6.68322620335964596377256305080, −5.88716538565891679878676892337, −4.43341110038143385096167907421, −3.60794019985913290145524014011, −2.86954270995766981613529818049, −0.52437466602640155689363788703,
0.52437466602640155689363788703, 2.86954270995766981613529818049, 3.60794019985913290145524014011, 4.43341110038143385096167907421, 5.88716538565891679878676892337, 6.68322620335964596377256305080, 8.020080629317086452733830379356, 9.188235332153868060901437519741, 10.48507419193051475897500424088, 11.75942563620851620754048103243
