| L(s) = 1 | − 0.725·2-s + 19.6·3-s − 31.4·4-s − 46.7·5-s − 14.2·6-s − 49·7-s + 46.0·8-s + 143.·9-s + 33.8·10-s − 618.·12-s + 650.·13-s + 35.5·14-s − 918.·15-s + 973.·16-s − 1.18e3·17-s − 103.·18-s + 1.56e3·19-s + 1.47e3·20-s − 962.·21-s − 1.10e3·23-s + 904.·24-s − 939.·25-s − 471.·26-s − 1.96e3·27-s + 1.54e3·28-s − 2.39e3·29-s + 666.·30-s + ⋯ |
| L(s) = 1 | − 0.128·2-s + 1.26·3-s − 0.983·4-s − 0.836·5-s − 0.161·6-s − 0.377·7-s + 0.254·8-s + 0.588·9-s + 0.107·10-s − 1.23·12-s + 1.06·13-s + 0.0484·14-s − 1.05·15-s + 0.950·16-s − 0.996·17-s − 0.0754·18-s + 0.994·19-s + 0.822·20-s − 0.476·21-s − 0.433·23-s + 0.320·24-s − 0.300·25-s − 0.136·26-s − 0.518·27-s + 0.371·28-s − 0.529·29-s + 0.135·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.546290383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.546290383\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 49T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.725T + 32T^{2} \) |
| 3 | \( 1 - 19.6T + 243T^{2} \) |
| 5 | \( 1 + 46.7T + 3.12e3T^{2} \) |
| 13 | \( 1 - 650.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.51e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 675.T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.29e4T + 8.58e9T^{2} \) |
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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
−9.267844495615412876094120599045, −8.547577301990521883958323233615, −8.066572470268800350162078373723, −7.22684309546307051823544621330, −5.95398862695451795222335247742, −4.72775627617332148058120873920, −3.66557485530379533053202569198, −3.41402249701064925120982412185, −1.91772140170548664357819384123, −0.53454319265707311897438087908,
0.53454319265707311897438087908, 1.91772140170548664357819384123, 3.41402249701064925120982412185, 3.66557485530379533053202569198, 4.72775627617332148058120873920, 5.95398862695451795222335247742, 7.22684309546307051823544621330, 8.066572470268800350162078373723, 8.547577301990521883958323233615, 9.267844495615412876094120599045
