| L(s) = 1 | + (14.6 − 14.6i)5-s + 24.0·7-s + (61.7 + 61.7i)11-s + (−37.5 − 37.5i)13-s − 96.8·17-s + (156. − 156. i)19-s + 959.·23-s + 198. i·25-s + (350. + 350. i)29-s − 237. i·31-s + (350. − 350. i)35-s + (−560. + 560. i)37-s + 1.80e3i·41-s + (−206. − 206. i)43-s − 1.59e3i·47-s + ⋯ |
| L(s) = 1 | + (0.584 − 0.584i)5-s + 0.490·7-s + (0.510 + 0.510i)11-s + (−0.222 − 0.222i)13-s − 0.335·17-s + (0.434 − 0.434i)19-s + 1.81·23-s + 0.317i·25-s + (0.416 + 0.416i)29-s − 0.247i·31-s + (0.286 − 0.286i)35-s + (−0.409 + 0.409i)37-s + 1.07i·41-s + (−0.111 − 0.111i)43-s − 0.724i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.713993988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.713993988\) |
| \(L(3)\) |
|
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 | \( 1 + (-14.6 + 14.6i)T - 625iT^{2} \) |
| 7 | \( 1 - 24.0T + 2.40e3T^{2} \) |
| 11 | \( 1 + (-61.7 - 61.7i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (37.5 + 37.5i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + 96.8T + 8.35e4T^{2} \) |
| 19 | \( 1 + (-156. + 156. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 - 959.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-350. - 350. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + 237. iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (560. - 560. i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.80e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (206. + 206. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 1.59e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (-2.23e3 + 2.23e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (-2.35e3 - 2.35e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (4.44e3 + 4.44e3i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (-3.99e3 + 3.99e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 4.92e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 2.65e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 8.79e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (228. - 228. i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.05e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 1.10e4T + 8.85e7T^{2} \) |
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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
−9.924327441344862170337136698240, −9.207772291128663618128623677573, −8.480182277319237647996532797949, −7.33460862623849972454434137922, −6.51432476478366718058420236222, −5.19647716562781372655614736571, −4.74944377276566201252936680343, −3.26222786126384819912075869853, −1.91138830889171710819604285635, −0.888878995745535133294642063202,
0.968668016675720085388044438759, 2.23806037971596534340952859991, 3.33448430876620190067631138894, 4.60431880319793617708919804315, 5.64617982346603616534432875204, 6.58267472549487240644742288249, 7.37310684619420269857663486574, 8.539161645959197597344004567989, 9.272157432224660292577270876813, 10.26110074518430238978750533286
