| L(s) = 1 | − 26.4·5-s − 31.3·7-s − 7.91·11-s + 252. i·13-s + 375.·17-s + (346. − 101. i)19-s + 472.·23-s + 73.1·25-s + 1.16e3i·29-s − 1.31e3i·31-s + 829.·35-s + 604. i·37-s − 3.16e3i·41-s − 1.81e3·43-s − 726.·47-s + ⋯ |
| L(s) = 1 | − 1.05·5-s − 0.640·7-s − 0.0654·11-s + 1.49i·13-s + 1.29·17-s + (0.960 − 0.279i)19-s + 0.894·23-s + 0.117·25-s + 1.38i·29-s − 1.36i·31-s + 0.677·35-s + 0.441i·37-s − 1.88i·41-s − 0.984·43-s − 0.328·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.960 + 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.06985881416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06985881416\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-346. + 101. i)T \) |
| good | 5 | \( 1 + 26.4T + 625T^{2} \) |
| 7 | \( 1 + 31.3T + 2.40e3T^{2} \) |
| 11 | \( 1 + 7.91T + 1.46e4T^{2} \) |
| 13 | \( 1 - 252. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 375.T + 8.35e4T^{2} \) |
| 23 | \( 1 - 472.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 1.16e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.31e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 604. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + 3.16e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.81e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 726.T + 4.87e6T^{2} \) |
| 53 | \( 1 - 5.49e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.99e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 3.50e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.07e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 2.67e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.25e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 8.21e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 1.14e4T + 4.74e7T^{2} \) |
| 89 | \( 1 - 3.16e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.51e4iT - 8.85e7T^{2} \) |
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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
−10.33947490681959063521927485002, −9.410119140209489213804784937347, −8.743330111516515021739476198300, −7.51985791365147694559973234879, −7.11873947525196887404152683163, −5.94105152432198027619050405891, −4.80253499464363536208142108043, −3.78842183789105967404715341123, −2.99432263219918034459766424233, −1.34764509804475578902324880477,
0.02007622063675672361139215385, 1.09137894408128707048216687701, 3.09692600179907481961897030266, 3.44665093456821811698590705052, 4.87464462840006662610278905621, 5.74897915044972096148154966294, 6.89150257497153843074062580909, 7.87568248640495759216507527992, 8.215955866294613617473900530977, 9.646781400732749347191577109477
