| L(s) = 1 | − 32·5-s + 49·7-s + 624·11-s − 708·13-s − 934·17-s + 1.85e3·19-s + 1.12e3·23-s − 2.10e3·25-s + 1.17e3·29-s + 2.90e3·31-s − 1.56e3·35-s − 1.24e4·37-s − 2.66e3·41-s − 7.14e3·43-s + 7.46e3·47-s + 2.40e3·49-s + 2.72e4·53-s − 1.99e4·55-s − 2.49e3·59-s − 1.10e4·61-s + 2.26e4·65-s + 3.97e4·67-s + 6.98e4·71-s + 1.64e4·73-s + 3.05e4·77-s + 7.83e4·79-s − 1.09e5·83-s + ⋯ |
| L(s) = 1 | − 0.572·5-s + 0.377·7-s + 1.55·11-s − 1.16·13-s − 0.783·17-s + 1.18·19-s + 0.441·23-s − 0.672·25-s + 0.259·29-s + 0.543·31-s − 0.216·35-s − 1.49·37-s − 0.247·41-s − 0.589·43-s + 0.493·47-s + 1/7·49-s + 1.33·53-s − 0.890·55-s − 0.0931·59-s − 0.381·61-s + 0.665·65-s + 1.08·67-s + 1.64·71-s + 0.361·73-s + 0.587·77-s + 1.41·79-s − 1.74·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.953801737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.953801737\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 32 T + p^{5} T^{2} \) |
| 11 | \( 1 - 624 T + p^{5} T^{2} \) |
| 13 | \( 1 + 708 T + p^{5} T^{2} \) |
| 17 | \( 1 + 934 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1858 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1120 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1174 T + p^{5} T^{2} \) |
| 31 | \( 1 - 2908 T + p^{5} T^{2} \) |
| 37 | \( 1 + 12462 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2662 T + p^{5} T^{2} \) |
| 43 | \( 1 + 7144 T + p^{5} T^{2} \) |
| 47 | \( 1 - 7468 T + p^{5} T^{2} \) |
| 53 | \( 1 - 27274 T + p^{5} T^{2} \) |
| 59 | \( 1 + 2490 T + p^{5} T^{2} \) |
| 61 | \( 1 + 11096 T + p^{5} T^{2} \) |
| 67 | \( 1 - 39756 T + p^{5} T^{2} \) |
| 71 | \( 1 - 69888 T + p^{5} T^{2} \) |
| 73 | \( 1 - 16450 T + p^{5} T^{2} \) |
| 79 | \( 1 - 78376 T + p^{5} T^{2} \) |
| 83 | \( 1 + 109818 T + p^{5} T^{2} \) |
| 89 | \( 1 - 56966 T + p^{5} T^{2} \) |
| 97 | \( 1 + 115946 T + p^{5} T^{2} \) |
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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
−10.02440519219289023206574267284, −9.235959471566366891246428240311, −8.366020585771334664440492652131, −7.30807815794708318943142528503, −6.67876116197668066964064355277, −5.31419901906406817022227176274, −4.37150364697475188852319536812, −3.40657596135082735799816824954, −1.99074323515603502775507146385, −0.71155608190346149353994044686,
0.71155608190346149353994044686, 1.99074323515603502775507146385, 3.40657596135082735799816824954, 4.37150364697475188852319536812, 5.31419901906406817022227176274, 6.67876116197668066964064355277, 7.30807815794708318943142528503, 8.366020585771334664440492652131, 9.235959471566366891246428240311, 10.02440519219289023206574267284
