| L(s) = 1 | + 30.1i·5-s + 52.3i·7-s − 90.0·11-s − 60.3i·13-s + 338·17-s + 6.92·19-s − 732. i·23-s − 287·25-s − 1.29e3i·29-s − 1.30e3i·31-s − 1.57e3·35-s − 241. i·37-s − 578·41-s + 2.02e3·43-s + 2.19e3i·47-s + ⋯ |
| L(s) = 1 | + 1.20i·5-s + 1.06i·7-s − 0.744·11-s − 0.357i·13-s + 1.16·17-s + 0.0191·19-s − 1.38i·23-s − 0.459·25-s − 1.54i·29-s − 1.36i·31-s − 1.28·35-s − 0.176i·37-s − 0.343·41-s + 1.09·43-s + 0.994i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.882966372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.882966372\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 30.1iT - 625T^{2} \) |
| 7 | \( 1 - 52.3iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 90.0T + 1.46e4T^{2} \) |
| 13 | \( 1 + 60.3iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 338T + 8.35e4T^{2} \) |
| 19 | \( 1 - 6.92T + 1.30e5T^{2} \) |
| 23 | \( 1 + 732. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.29e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.30e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 241. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 578T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.02e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.19e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.44e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 1.19e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 6.40e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 8.26e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 4.28e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 8.73e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.12e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.31e4T + 4.74e7T^{2} \) |
| 89 | \( 1 + 910T + 6.27e7T^{2} \) |
| 97 | \( 1 - 5.42e3T + 8.85e7T^{2} \) |
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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
−9.424332799863417280140971221189, −8.164479743490122971276960061213, −7.78804796839110621378566200477, −6.61674474589300111335588745678, −5.94686567808535132700306766262, −5.15311585862818439236687723050, −3.84302301657577826974050465810, −2.71984505465308676734964285526, −2.32645304898405767077708900964, −0.49336405602045809382064856743,
0.862731231821676076865272957900, 1.51265736608282480410452283864, 3.15158485786315712760027768100, 4.06327483745814172307957654558, 5.06481134329232952090541124165, 5.55800104919344542821395737093, 6.98887869751738839258999824203, 7.57277666712569761963586487521, 8.462973402936904680033032568112, 9.183753411862665859634823558426
