| L(s) = 1 | + 96·5-s − 49·7-s − 720·11-s + 572·13-s − 1.25e3·17-s + 94·19-s + 96·23-s + 6.09e3·25-s + 4.37e3·29-s + 6.24e3·31-s − 4.70e3·35-s − 1.07e4·37-s − 1.20e4·41-s + 9.16e3·43-s − 2.58e4·47-s + 2.40e3·49-s − 1.01e3·53-s − 6.91e4·55-s + 1.24e3·59-s + 7.59e3·61-s + 5.49e4·65-s − 4.11e4·67-s − 3.76e4·71-s − 1.34e4·73-s + 3.52e4·77-s − 6.24e3·79-s − 2.52e4·83-s + ⋯ |
| L(s) = 1 | + 1.71·5-s − 0.377·7-s − 1.79·11-s + 0.938·13-s − 1.05·17-s + 0.0597·19-s + 0.0378·23-s + 1.94·25-s + 0.965·29-s + 1.16·31-s − 0.649·35-s − 1.29·37-s − 1.11·41-s + 0.755·43-s − 1.70·47-s + 1/7·49-s − 0.0495·53-s − 3.08·55-s + 0.0464·59-s + 0.261·61-s + 1.61·65-s − 1.11·67-s − 0.885·71-s − 0.295·73-s + 0.678·77-s − 0.112·79-s − 0.402·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 5 | \( 1 - 96 T + p^{5} T^{2} \) |
| 11 | \( 1 + 720 T + p^{5} T^{2} \) |
| 13 | \( 1 - 44 p T + p^{5} T^{2} \) |
| 17 | \( 1 + 1254 T + p^{5} T^{2} \) |
| 19 | \( 1 - 94 T + p^{5} T^{2} \) |
| 23 | \( 1 - 96 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4374 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6244 T + p^{5} T^{2} \) |
| 37 | \( 1 + 10798 T + p^{5} T^{2} \) |
| 41 | \( 1 + 12006 T + p^{5} T^{2} \) |
| 43 | \( 1 - 9160 T + p^{5} T^{2} \) |
| 47 | \( 1 + 25836 T + p^{5} T^{2} \) |
| 53 | \( 1 + 1014 T + p^{5} T^{2} \) |
| 59 | \( 1 - 1242 T + p^{5} T^{2} \) |
| 61 | \( 1 - 7592 T + p^{5} T^{2} \) |
| 67 | \( 1 + 41132 T + p^{5} T^{2} \) |
| 71 | \( 1 + 37632 T + p^{5} T^{2} \) |
| 73 | \( 1 + 13438 T + p^{5} T^{2} \) |
| 79 | \( 1 + 6248 T + p^{5} T^{2} \) |
| 83 | \( 1 + 25254 T + p^{5} T^{2} \) |
| 89 | \( 1 - 45126 T + p^{5} T^{2} \) |
| 97 | \( 1 - 107222 T + p^{5} T^{2} \) |
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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
−8.830300181827958420312820179025, −8.194848342018045133090884719955, −6.86718655212939747540735479240, −6.22912996534650741085379067927, −5.43415484588931354688228223640, −4.68927453815794082093441911485, −3.09321245068417926461634497036, −2.38254420687638275301604227673, −1.39523458724038252301449199880, 0,
1.39523458724038252301449199880, 2.38254420687638275301604227673, 3.09321245068417926461634497036, 4.68927453815794082093441911485, 5.43415484588931354688228223640, 6.22912996534650741085379067927, 6.86718655212939747540735479240, 8.194848342018045133090884719955, 8.830300181827958420312820179025
