| L(s) = 1 | − 4·2-s − 30·3-s + 16·4-s + 25·5-s + 120·6-s − 49·7-s − 64·8-s + 657·9-s − 100·10-s + 121·11-s − 480·12-s + 408·13-s + 196·14-s − 750·15-s + 256·16-s − 252·17-s − 2.62e3·18-s − 372·19-s + 400·20-s + 1.47e3·21-s − 484·22-s − 2.31e3·23-s + 1.92e3·24-s + 625·25-s − 1.63e3·26-s − 1.24e4·27-s − 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.92·3-s + 1/2·4-s + 0.447·5-s + 1.36·6-s − 0.377·7-s − 0.353·8-s + 2.70·9-s − 0.316·10-s + 0.301·11-s − 0.962·12-s + 0.669·13-s + 0.267·14-s − 0.860·15-s + 1/4·16-s − 0.211·17-s − 1.91·18-s − 0.236·19-s + 0.223·20-s + 0.727·21-s − 0.213·22-s − 0.912·23-s + 0.680·24-s + 1/5·25-s − 0.473·26-s − 3.27·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| 11 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 + 10 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 408 T + p^{5} T^{2} \) |
| 17 | \( 1 + 252 T + p^{5} T^{2} \) |
| 19 | \( 1 + 372 T + p^{5} T^{2} \) |
| 23 | \( 1 + 2314 T + p^{5} T^{2} \) |
| 29 | \( 1 - 3266 T + p^{5} T^{2} \) |
| 31 | \( 1 + 1808 T + p^{5} T^{2} \) |
| 37 | \( 1 + 1542 T + p^{5} T^{2} \) |
| 41 | \( 1 + 10570 T + p^{5} T^{2} \) |
| 43 | \( 1 + 6564 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1722 T + p^{5} T^{2} \) |
| 53 | \( 1 + 15006 T + p^{5} T^{2} \) |
| 59 | \( 1 - 49148 T + p^{5} T^{2} \) |
| 61 | \( 1 - 5666 T + p^{5} T^{2} \) |
| 67 | \( 1 + 26318 T + p^{5} T^{2} \) |
| 71 | \( 1 + 1612 T + p^{5} T^{2} \) |
| 73 | \( 1 - 63332 T + p^{5} T^{2} \) |
| 79 | \( 1 + 40360 T + p^{5} T^{2} \) |
| 83 | \( 1 - 63972 T + p^{5} T^{2} \) |
| 89 | \( 1 - 85830 T + p^{5} T^{2} \) |
| 97 | \( 1 - 13526 T + p^{5} T^{2} \) |
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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
−9.443068111517155229561292286828, −8.303880195216979401020191639860, −7.06090299899621354593704642134, −6.44166335178539658206510812060, −5.85822673341275843134829929854, −4.90601287767774988106247738509, −3.74380643589001830338277365980, −1.92787873241566742692486044924, −0.956004027162983843488984559054, 0,
0.956004027162983843488984559054, 1.92787873241566742692486044924, 3.74380643589001830338277365980, 4.90601287767774988106247738509, 5.85822673341275843134829929854, 6.44166335178539658206510812060, 7.06090299899621354593704642134, 8.303880195216979401020191639860, 9.443068111517155229561292286828
