| L(s) = 1 | + (−5.54 + 5.54i)3-s + (21.7 − 21.7i)5-s + 6.62·7-s + 19.6i·9-s + (90.9 + 90.9i)11-s + (221. + 221. i)13-s + 240. i·15-s − 132.·17-s + (402. − 402. i)19-s + (−36.6 + 36.6i)21-s − 27.5·23-s − 320. i·25-s + (−557. − 557. i)27-s + (174. + 174. i)29-s + 1.08e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.615 + 0.615i)3-s + (0.869 − 0.869i)5-s + 0.135·7-s + 0.242i·9-s + (0.752 + 0.752i)11-s + (1.31 + 1.31i)13-s + 1.07i·15-s − 0.458·17-s + (1.11 − 1.11i)19-s + (−0.0831 + 0.0831i)21-s − 0.0519·23-s − 0.512i·25-s + (−0.764 − 0.764i)27-s + (0.207 + 0.207i)29-s + 1.12i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.792 - 0.609i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.51630 + 0.515283i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51630 + 0.515283i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (5.54 - 5.54i)T - 81iT^{2} \) |
| 5 | \( 1 + (-21.7 + 21.7i)T - 625iT^{2} \) |
| 7 | \( 1 - 6.62T + 2.40e3T^{2} \) |
| 11 | \( 1 + (-90.9 - 90.9i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (-221. - 221. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + 132.T + 8.35e4T^{2} \) |
| 19 | \( 1 + (-402. + 402. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + 27.5T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-174. - 174. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 - 1.08e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (-553. + 553. i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 + 1.80e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (17.8 + 17.8i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + 2.26e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (822. - 822. i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (-972. - 972. i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (2.05e3 + 2.05e3i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (4.61e3 - 4.61e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 3.10e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 723. iT - 2.83e7T^{2} \) |
| 79 | \( 1 - 3.41e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-161. + 161. i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 1.46e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 8.26e3T + 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.08301660360420082542174123430, −13.31315313343587481507764876742, −11.89507736346306994595756376413, −10.93665181349684636028948799348, −9.571249572713310889546648027620, −8.836280546976596070506473291044, −6.78030877942215432409665643464, −5.37610279586606145750541596226, −4.33277085293575615610386874334, −1.57512975353306737512138172494,
1.17408173312571183265939781091, 3.32075078808444484223354400864, 5.89954202878228836730733786217, 6.32984520792163636782156618357, 7.966455322283301743490047065929, 9.576377578719218582629805651659, 10.82270898520987772165194093770, 11.69439529468234509127812884065, 13.05912608150907783415471244609, 13.92103818052037797198048528451
