| L(s) = 1 | − 6.72·5-s − 35.8i·7-s + (−120. + 4.60i)11-s − 138. i·13-s + 364. i·17-s − 570. i·19-s − 70.8·23-s − 579.·25-s + 317. i·29-s + 1.15e3·31-s + 241. i·35-s − 1.36e3·37-s − 1.33e3i·41-s + 2.15e3i·43-s − 1.41e3·47-s + ⋯ |
| L(s) = 1 | − 0.268·5-s − 0.731i·7-s + (−0.999 + 0.0380i)11-s − 0.817i·13-s + 1.26i·17-s − 1.58i·19-s − 0.133·23-s − 0.927·25-s + 0.377i·29-s + 1.20·31-s + 0.196i·35-s − 0.994·37-s − 0.795i·41-s + 1.16i·43-s − 0.641·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0380 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.0380 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6958808676\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6958808676\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (120. - 4.60i)T \) |
| good | 5 | \( 1 + 6.72T + 625T^{2} \) |
| 7 | \( 1 + 35.8iT - 2.40e3T^{2} \) |
| 13 | \( 1 + 138. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 364. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 570. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 70.8T + 2.79e5T^{2} \) |
| 29 | \( 1 - 317. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.15e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.36e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.33e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.15e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.41e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 3.01e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 1.66e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 5.39e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 588.T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.02e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 2.25e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 6.31e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 777. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.64e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 6.32e3T + 8.85e7T^{2} \) |
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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
−10.28173313652611955405585563741, −8.985200131134480010437691144195, −8.092329750534653155879525389859, −7.48059579908791229744461815631, −6.49751328030251526944528889756, −5.43747972378662918958698469558, −4.50774245779250294743163261418, −3.48746783013574319734696642086, −2.40363678411042546042036414100, −0.908533433281623359414959243016,
0.18769041556303083932745266263, 1.82985536802950418326772243343, 2.82099892566755171149212543134, 3.99254892960838460952704439687, 5.09341692355022464689747922206, 5.86102043337062420172150150818, 6.92147984702020120760140878692, 7.894832433662644323967375132947, 8.515970329557976299040557441830, 9.594755560881205205829103847428
