| L(s) = 1 | + (−0.951 + 0.309i)2-s + (−0.909 − 1.25i)3-s + (0.809 − 0.587i)4-s + (2.02 + 0.940i)5-s + (1.25 + 0.909i)6-s + i·7-s + (−0.587 + 0.809i)8-s + (0.186 − 0.574i)9-s + (−2.21 − 0.267i)10-s + (1.82 + 5.60i)11-s + (−1.47 − 0.478i)12-s + (−1.64 − 0.533i)13-s + (−0.309 − 0.951i)14-s + (−0.667 − 3.39i)15-s + (0.309 − 0.951i)16-s + (1.58 − 2.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (−0.525 − 0.722i)3-s + (0.404 − 0.293i)4-s + (0.907 + 0.420i)5-s + (0.511 + 0.371i)6-s + 0.377i·7-s + (−0.207 + 0.286i)8-s + (0.0622 − 0.191i)9-s + (−0.702 − 0.0847i)10-s + (0.548 + 1.68i)11-s + (−0.424 − 0.138i)12-s + (−0.455 − 0.147i)13-s + (−0.0825 − 0.254i)14-s + (−0.172 − 0.876i)15-s + (0.0772 − 0.237i)16-s + (0.384 − 0.529i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.136i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00342 + 0.0687457i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00342 + 0.0687457i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (-2.02 - 0.940i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (0.909 + 1.25i)T + (-0.927 + 2.85i)T^{2} \) |
| 11 | \( 1 + (-1.82 - 5.60i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.64 + 0.533i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.58 + 2.18i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.09 - 0.799i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-5.13 + 1.66i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.28 + 3.84i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.10 - 4.43i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.163 - 0.0531i)T + (29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.53 + 4.72i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.06iT - 43T^{2} \) |
| 47 | \( 1 + (0.297 + 0.409i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (1.68 + 2.31i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.0158 - 0.0486i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.778 + 2.39i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (6.01 - 8.28i)T + (-20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (8.91 - 6.47i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (10.9 - 3.57i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.646 - 0.469i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.40 + 8.81i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (4.47 + 13.7i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (0.0557 + 0.0766i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−11.66216347906082052086144568100, −10.27878529537589337987380717802, −9.756956875798757378838747295688, −8.869458161786091906194138566313, −7.35464708134007796802808353663, −6.88059663346359904500870994767, −6.00148640943855787744406607867, −4.84803713786832265836882421016, −2.65077961994007158931262693644, −1.37963617234155408177951453010,
1.15369975341992723774755539319, 3.02504046883366286159924974740, 4.52404724480517903466259872116, 5.63172457191508945528306030166, 6.52048328526257445447789781516, 7.951667194152275765779688958559, 8.928892140976277516958330652447, 9.698429108777727059980165228267, 10.53270511449941033114558070005, 11.13230135549702815812684813961
