| L(s) = 1 | − 64·2-s + 4.09e3·4-s + 2.35e4·5-s − 2.62e5·8-s + 5.31e5·9-s − 1.50e6·10-s + 6.91e6·13-s + 1.67e7·16-s − 4.72e7·17-s − 3.40e7·18-s + 9.62e7·20-s + 3.08e8·25-s − 4.42e8·26-s − 1.73e8·29-s − 1.07e9·32-s + 3.02e9·34-s + 2.17e9·36-s − 2.05e9·37-s − 6.16e9·40-s − 2.28e9·41-s + 1.24e10·45-s + 1.38e10·49-s − 1.97e10·50-s + 2.83e10·52-s − 4.34e10·53-s + 1.11e10·58-s − 4.78e10·61-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 1.50·5-s − 8-s + 9-s − 1.50·10-s + 1.43·13-s + 16-s − 1.95·17-s − 18-s + 1.50·20-s + 1.26·25-s − 1.43·26-s − 0.291·29-s − 32-s + 1.95·34-s + 36-s − 0.799·37-s − 1.50·40-s − 0.481·41-s + 1.50·45-s + 49-s − 1.26·50-s + 1.43·52-s − 1.96·53-s + 0.291·58-s − 0.928·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.215168254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.215168254\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{6} T \) |
| good | 3 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 5 | \( 1 - 23506 T + p^{12} T^{2} \) |
| 7 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 11 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 13 | \( 1 - 6911282 T + p^{12} T^{2} \) |
| 17 | \( 1 + 47295038 T + p^{12} T^{2} \) |
| 19 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 23 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 29 | \( 1 + 173439758 T + p^{12} T^{2} \) |
| 31 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 37 | \( 1 + 2050092718 T + p^{12} T^{2} \) |
| 41 | \( 1 + 2285065118 T + p^{12} T^{2} \) |
| 43 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 47 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 53 | \( 1 + 43462597358 T + p^{12} T^{2} \) |
| 59 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 61 | \( 1 + 47844884878 T + p^{12} T^{2} \) |
| 67 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 71 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 73 | \( 1 + 119852347678 T + p^{12} T^{2} \) |
| 79 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 83 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 89 | \( 1 - 907573615522 T + p^{12} T^{2} \) |
| 97 | \( 1 - 502341690242 T + p^{12} T^{2} \) |
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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
−21.74181247581787008778530113983, −20.56768855042740253895318316155, −18.51460181035323228390667973884, −17.51327125303811760908080700334, −15.75162371202038124789399710352, −13.32375910129298937575455768067, −10.65301596592464524474136750475, −9.099703523665283906261706405805, −6.44774866235029399555885813687, −1.72643016907722640515214317980,
1.72643016907722640515214317980, 6.44774866235029399555885813687, 9.099703523665283906261706405805, 10.65301596592464524474136750475, 13.32375910129298937575455768067, 15.75162371202038124789399710352, 17.51327125303811760908080700334, 18.51460181035323228390667973884, 20.56768855042740253895318316155, 21.74181247581787008778530113983
