| L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.669 + 0.743i)3-s + (−0.809 − 0.587i)4-s + (−0.5 + 0.866i)5-s + (−0.499 − 0.866i)6-s + (0.167 − 1.59i)7-s + (0.809 − 0.587i)8-s + (−0.104 − 0.994i)9-s + (−0.669 − 0.743i)10-s + (−0.863 + 0.384i)11-s + (0.978 − 0.207i)12-s + (−3.71 − 0.790i)13-s + (1.46 + 0.651i)14-s + (−0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (0.653 + 0.291i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.386 + 0.429i)3-s + (−0.404 − 0.293i)4-s + (−0.223 + 0.387i)5-s + (−0.204 − 0.353i)6-s + (0.0632 − 0.601i)7-s + (0.286 − 0.207i)8-s + (−0.0348 − 0.331i)9-s + (−0.211 − 0.235i)10-s + (−0.260 + 0.115i)11-s + (0.282 − 0.0600i)12-s + (−1.03 − 0.219i)13-s + (0.390 + 0.174i)14-s + (−0.0797 − 0.245i)15-s + (0.0772 + 0.237i)16-s + (0.158 + 0.0705i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.867956 - 0.0439262i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.867956 - 0.0439262i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (2.90 + 4.75i)T \) |
| good | 7 | \( 1 + (-0.167 + 1.59i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (0.863 - 0.384i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (3.71 + 0.790i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-0.653 - 0.291i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-1.84 + 0.391i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-6.29 + 4.57i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.40 + 7.39i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (4.00 + 6.94i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.31 - 5.90i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (2.85 - 0.607i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-3.67 - 11.3i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.03 + 9.80i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-4.10 + 4.56i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + (4.99 - 8.64i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.434 + 4.13i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-3.84 + 1.71i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-4.88 - 2.17i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (4.39 + 4.88i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-11.6 - 8.46i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (9.01 + 6.55i)T + (29.9 + 92.2i)T^{2} \) |
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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
−9.965565253571850386819018189481, −9.373582075336402339985678235189, −8.202530165175020464062310138354, −7.42809553472114639799902511100, −6.78165644696569739083669821912, −5.73194024016097105751801203092, −4.83139259450223943837891966068, −4.01604536519252915209085233215, −2.64638388278953174984857557998, −0.54092574945721437803512965017,
1.17107829249362790747789883623, 2.44466498996768574752385113286, 3.53922240289186041287091001305, 5.06011287324502190873463291332, 5.33287205315709918945273142040, 6.90079416764343377816329305795, 7.52754636675141674282591970860, 8.651876464200616718305373225989, 9.158630517125205663617825479050, 10.20153753557553573152643438968
