| 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
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1288659672\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1288659672\) |
| \(L(3)\) |
|
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
−10.36687847646170146049958640284, −9.427406968358203152921229317698, −8.766976563245959252882533380576, −7.61551619367457361715769102903, −6.64841350300889168624061054577, −5.79344503452082563122291882460, −5.08799444869320925651908056971, −3.60002135238574081511578157588, −2.64034647595060750686358373214, −1.42804450858112871875416290481,
0.02776892251277931433334405353, 1.69252892992644233679126638298, 2.52495756711264272626473579715, 3.93600288615400324280561215901, 4.91202310874430338472096997397, 6.29541207924003314999300316396, 6.42094532680789604888016529014, 7.72763334849512831695536558676, 8.882746035736607168415125475687, 9.704985230907314990939743938234
