| L(s) = 1 | − 4·2-s + 3-s + 16·4-s − 51·5-s − 4·6-s − 166·7-s − 64·8-s − 242·9-s + 204·10-s − 121·11-s + 16·12-s + 692·13-s + 664·14-s − 51·15-s + 256·16-s − 738·17-s + 968·18-s + 1.42e3·19-s − 816·20-s − 166·21-s + 484·22-s − 1.77e3·23-s − 64·24-s − 524·25-s − 2.76e3·26-s − 485·27-s − 2.65e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.0641·3-s + 1/2·4-s − 0.912·5-s − 0.0453·6-s − 1.28·7-s − 0.353·8-s − 0.995·9-s + 0.645·10-s − 0.301·11-s + 0.0320·12-s + 1.13·13-s + 0.905·14-s − 0.0585·15-s + 1/4·16-s − 0.619·17-s + 0.704·18-s + 0.904·19-s − 0.456·20-s − 0.0821·21-s + 0.213·22-s − 0.701·23-s − 0.0226·24-s − 0.167·25-s − 0.803·26-s − 0.128·27-s − 0.640·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{2} T \) |
| 11 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - T + p^{5} T^{2} \) |
| 5 | \( 1 + 51 T + p^{5} T^{2} \) |
| 7 | \( 1 + 166 T + p^{5} T^{2} \) |
| 13 | \( 1 - 692 T + p^{5} T^{2} \) |
| 17 | \( 1 + 738 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1424 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1779 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2064 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6245 T + p^{5} T^{2} \) |
| 37 | \( 1 + 14785 T + p^{5} T^{2} \) |
| 41 | \( 1 - 5304 T + p^{5} T^{2} \) |
| 43 | \( 1 - 17798 T + p^{5} T^{2} \) |
| 47 | \( 1 + 17184 T + p^{5} T^{2} \) |
| 53 | \( 1 + 30726 T + p^{5} T^{2} \) |
| 59 | \( 1 + 34989 T + p^{5} T^{2} \) |
| 61 | \( 1 + 45940 T + p^{5} T^{2} \) |
| 67 | \( 1 - 25343 T + p^{5} T^{2} \) |
| 71 | \( 1 - 13311 T + p^{5} T^{2} \) |
| 73 | \( 1 + 53260 T + p^{5} T^{2} \) |
| 79 | \( 1 - 77234 T + p^{5} T^{2} \) |
| 83 | \( 1 - 55014 T + p^{5} T^{2} \) |
| 89 | \( 1 - 125415 T + p^{5} T^{2} \) |
| 97 | \( 1 + 88807 T + p^{5} T^{2} \) |
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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
−16.13704085164519287550561730610, −15.62200030344491510526142876105, −13.71930333705818205251192437822, −12.11092717465564811607596492324, −10.89267977723568722889815913352, −9.270707670509439734046639466414, −7.960663838561403831392220075405, −6.24607614791838155446468459828, −3.31591632980037824949397176501, 0,
3.31591632980037824949397176501, 6.24607614791838155446468459828, 7.960663838561403831392220075405, 9.270707670509439734046639466414, 10.89267977723568722889815913352, 12.11092717465564811607596492324, 13.71930333705818205251192437822, 15.62200030344491510526142876105, 16.13704085164519287550561730610
