| L(s) = 1 | + (4.36 + 7.56i)5-s + (−14.4 − 11.5i)7-s + (7.60 + 4.39i)11-s + 11.8i·13-s + (−22.2 + 38.6i)17-s + (−10.0 + 5.82i)19-s + (123. − 71.3i)23-s + (24.3 − 42.1i)25-s + 234. i·29-s + (−252. − 145. i)31-s + (23.9 − 160. i)35-s + (44.4 + 76.9i)37-s − 145.·41-s − 144.·43-s + (−120. − 208. i)47-s + ⋯ |
| L(s) = 1 | + (0.390 + 0.676i)5-s + (−0.782 − 0.622i)7-s + (0.208 + 0.120i)11-s + 0.252i·13-s + (−0.318 + 0.550i)17-s + (−0.121 + 0.0703i)19-s + (1.11 − 0.646i)23-s + (0.194 − 0.337i)25-s + 1.49i·29-s + (−1.46 − 0.845i)31-s + (0.115 − 0.772i)35-s + (0.197 + 0.342i)37-s − 0.555·41-s − 0.512·43-s + (−0.372 − 0.646i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1345044190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1345044190\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (14.4 + 11.5i)T \) |
| good | 5 | \( 1 + (-4.36 - 7.56i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-7.60 - 4.39i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 11.8iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (22.2 - 38.6i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (10.0 - 5.82i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-123. + 71.3i)T + (6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 234. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (252. + 145. i)T + (1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-44.4 - 76.9i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 145.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 144.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (120. + 208. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (263. + 152. i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-3.54 + 6.13i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (149. - 86.4i)T + (1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (243. - 421. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 653. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (99.0 + 57.1i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-147. - 255. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 877.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (710. + 1.23e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 738. iT - 9.12e5T^{2} \) |
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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
−9.286301324037971387221995651728, −8.517996118922915620793298740810, −7.22869557802835324484400610832, −6.77117255156128351831407673718, −5.99219005665301686308675543419, −4.78364290150860825743546701422, −3.70558957499766215888101304995, −2.85237933298889551147025396016, −1.57785035943779076673226656169, −0.03287082706919651521109232894,
1.34033405742634834525018539231, 2.62044428463457443046337955007, 3.60842085097019168316040535191, 4.91307665520163826540044394366, 5.58668047057802903760283017977, 6.49113801261193029411678205960, 7.38330722610193071552565621820, 8.498019410790884169961210829860, 9.270364061497370902868194737440, 9.592369945905319473011079421556
