| 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
−10.20153753557553573152643438968, −9.158630517125205663617825479050, −8.651876464200616718305373225989, −7.52754636675141674282591970860, −6.90079416764343377816329305795, −5.33287205315709918945273142040, −5.06011287324502190873463291332, −3.53922240289186041287091001305, −2.44466498996768574752385113286, −1.17107829249362790747789883623,
0.54092574945721437803512965017, 2.64638388278953174984857557998, 4.01604536519252915209085233215, 4.83139259450223943837891966068, 5.73194024016097105751801203092, 6.78165644696569739083669821912, 7.42809553472114639799902511100, 8.202530165175020464062310138354, 9.373582075336402339985678235189, 9.965565253571850386819018189481
