| L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.671 + 1.59i)3-s + (−0.939 − 0.342i)4-s + (−1.20 + 1.01i)5-s + (−1.68 + 0.384i)6-s + (−0.939 + 0.342i)7-s + (0.5 − 0.866i)8-s + (−2.09 + 2.14i)9-s + (−0.786 − 1.36i)10-s + (−3.22 − 2.70i)11-s + (−0.0851 − 1.72i)12-s + (0.953 + 5.40i)13-s + (−0.173 − 0.984i)14-s + (−2.42 − 1.24i)15-s + (0.766 + 0.642i)16-s + (−1.83 − 3.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.387 + 0.921i)3-s + (−0.469 − 0.171i)4-s + (−0.539 + 0.452i)5-s + (−0.689 + 0.156i)6-s + (−0.355 + 0.129i)7-s + (0.176 − 0.306i)8-s + (−0.699 + 0.714i)9-s + (−0.248 − 0.430i)10-s + (−0.971 − 0.815i)11-s + (−0.0245 − 0.499i)12-s + (0.264 + 1.49i)13-s + (−0.0464 − 0.263i)14-s + (−0.625 − 0.321i)15-s + (0.191 + 0.160i)16-s + (−0.444 − 0.769i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 + 0.260i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.106332 - 0.801343i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.106332 - 0.801343i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (-0.671 - 1.59i)T \) |
| 7 | \( 1 + (0.939 - 0.342i)T \) |
| good | 5 | \( 1 + (1.20 - 1.01i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (3.22 + 2.70i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.953 - 5.40i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (1.83 + 3.17i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.870 - 1.50i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.64 - 0.598i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.0573 - 0.325i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-5.96 - 2.17i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (4.26 + 7.38i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.82 - 10.3i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-6.25 - 5.25i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-3.41 + 1.24i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + (0.221 - 0.185i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-3.25 + 1.18i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.858 - 4.87i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.47 - 11.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.54 - 2.67i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.49 - 14.1i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.03 + 11.5i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (6.42 - 11.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.42 - 7.07i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.46481619009819330675843778731, −10.95523833649505267623418043294, −9.822444380244974729984528558672, −9.046897012018511449357581491441, −8.238014900834332581706089732019, −7.24319218329748747159008973073, −6.14616952941016713854466195872, −4.98418179881524276943374127615, −3.94466895283429149818242384763, −2.79908904948420418729236269789,
0.52513612965546943409618719882, 2.27320023514776860614435081154, 3.38813759591512670475079676438, 4.75168081701254214912737094073, 6.08181723321213333346071441221, 7.41723783054942547588960651093, 8.128298868463695423374019906710, 8.848992817769952124591226382959, 10.12085406585184690208992104405, 10.82403700852994689450517710830
