| L(s) = 1 | + 3.65·5-s − 9.65i·7-s + 18.4i·11-s − 11.6·13-s − 9.31·17-s + 15.1i·19-s + 22.3i·23-s − 11.6·25-s − 28.3·29-s + 45.2i·31-s − 35.3i·35-s + 49.5·37-s + 20.6·41-s + 46.0i·43-s − 12.6i·47-s + ⋯ |
| L(s) = 1 | + 0.731·5-s − 1.37i·7-s + 1.68i·11-s − 0.896·13-s − 0.547·17-s + 0.798i·19-s + 0.971i·23-s − 0.465·25-s − 0.977·29-s + 1.45i·31-s − 1.00i·35-s + 1.34·37-s + 0.503·41-s + 1.07i·43-s − 0.269i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.402908456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402908456\) |
| \(L(2)\) |
|
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 | \( 1 - 3.65T + 25T^{2} \) |
| 7 | \( 1 + 9.65iT - 49T^{2} \) |
| 11 | \( 1 - 18.4iT - 121T^{2} \) |
| 13 | \( 1 + 11.6T + 169T^{2} \) |
| 17 | \( 1 + 9.31T + 289T^{2} \) |
| 19 | \( 1 - 15.1iT - 361T^{2} \) |
| 23 | \( 1 - 22.3iT - 529T^{2} \) |
| 29 | \( 1 + 28.3T + 841T^{2} \) |
| 31 | \( 1 - 45.2iT - 961T^{2} \) |
| 37 | \( 1 - 49.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 20.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 46.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 12.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 27.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 9.11iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 113.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 45.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 16.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 11.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 70.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 94.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 110.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 25.4T + 9.40e3T^{2} \) |
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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
−9.819934738034420195951653044479, −9.368020047088436527068075153512, −7.907460532054480320381407826374, −7.30693287650460260643323562743, −6.67569389270768215439594688134, −5.49233862532290499871106146429, −4.58308482587066920106240595792, −3.81359042992171121528247121536, −2.32172169616497785474686100665, −1.37954111021008924505252364283,
0.40523535006971082673779612066, 2.23725405442507312031015686446, 2.73210282463609309240009677697, 4.18136666892342915375042209231, 5.49182804523291770368751606873, 5.79390213522096221376408592971, 6.71075810099601137739577856587, 7.927594040388835819854974884745, 8.761932954019221096097810644133, 9.270181180923825627737128574291
