| 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 + (0.0963 + 0.916i)7-s + (0.809 + 0.587i)8-s + (−0.104 + 0.994i)9-s + (−0.669 + 0.743i)10-s + (0.338 + 0.150i)11-s + (0.978 + 0.207i)12-s + (4.54 − 0.965i)13-s + (0.841 − 0.374i)14-s + (−0.309 + 0.951i)15-s + (0.309 − 0.951i)16-s + (1.57 − 0.699i)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 + (0.0364 + 0.346i)7-s + (0.286 + 0.207i)8-s + (−0.0348 + 0.331i)9-s + (−0.211 + 0.235i)10-s + (0.102 + 0.0454i)11-s + (0.282 + 0.0600i)12-s + (1.26 − 0.267i)13-s + (0.224 − 0.100i)14-s + (−0.0797 + 0.245i)15-s + (0.0772 − 0.237i)16-s + (0.380 − 0.169i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.384 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.620024 - 0.929881i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.620024 - 0.929881i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.80 + 4.81i)T \) |
| good | 7 | \( 1 + (-0.0963 - 0.916i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (-0.338 - 0.150i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-4.54 + 0.965i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-1.57 + 0.699i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-3.27 - 0.695i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (0.334 + 0.243i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.86 + 8.82i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (0.498 - 0.862i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.000431 - 0.000478i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-7.35 - 1.56i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (1.72 - 5.31i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.16 + 11.1i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (6.81 + 7.57i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + 5.50T + 61T^{2} \) |
| 67 | \( 1 + (-0.427 - 0.741i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.00473 + 0.0450i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (2.39 + 1.06i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-0.432 + 0.192i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-5.00 + 5.55i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-11.7 + 8.54i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (5.40 - 3.93i)T + (29.9 - 92.2i)T^{2} \) |
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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
−9.786505979269545776110806372453, −9.077669741811684693674085108196, −8.117908209583395523949445980913, −7.55462164611500554847199121732, −6.19034807492091015732234876996, −5.50725888662278769419040798736, −4.31786215545623711775841656917, −3.31537852927515499131901263681, −1.95390114600473227539291390670, −0.72167978133236471494588615019,
1.23385628161860512173668130095, 3.30417764047945257142363796423, 4.11510415357629010108364792471, 5.27227854983286504540539274552, 6.03739203790009986094700667287, 6.97132516242936049786812631456, 7.64051075716448797126494261074, 8.788576663631319478373473013093, 9.286332321404588692156862665580, 10.57700280864313201000880503359
