| L(s) = 1 | + (−1.28 − 0.587i)2-s + (−1.02 + 3.48i)3-s + (1.30 + 1.51i)4-s + (−2.37 + 4.40i)5-s + (3.36 − 3.87i)6-s + (−1.08 + 7.54i)7-s + (−0.796 − 2.71i)8-s + (−3.51 − 2.25i)9-s + (5.63 − 4.26i)10-s + (−6.50 + 2.97i)11-s + (−6.60 + 3.01i)12-s + (7.13 − 1.02i)13-s + (5.82 − 9.06i)14-s + (−12.9 − 12.7i)15-s + (−0.569 + 3.95i)16-s + (5.52 − 6.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.643 − 0.293i)2-s + (−0.340 + 1.16i)3-s + (0.327 + 0.377i)4-s + (−0.474 + 0.880i)5-s + (0.560 − 0.646i)6-s + (−0.154 + 1.07i)7-s + (−0.0996 − 0.339i)8-s + (−0.390 − 0.250i)9-s + (0.563 − 0.426i)10-s + (−0.591 + 0.270i)11-s + (−0.550 + 0.251i)12-s + (0.548 − 0.0788i)13-s + (0.416 − 0.647i)14-s + (−0.860 − 0.850i)15-s + (−0.0355 + 0.247i)16-s + (0.324 − 0.374i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.982 + 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0538072 - 0.567481i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0538072 - 0.567481i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.28 + 0.587i)T \) |
| 5 | \( 1 + (2.37 - 4.40i)T \) |
| 23 | \( 1 + (19.1 + 12.7i)T \) |
| good | 3 | \( 1 + (1.02 - 3.48i)T + (-7.57 - 4.86i)T^{2} \) |
| 7 | \( 1 + (1.08 - 7.54i)T + (-47.0 - 13.8i)T^{2} \) |
| 11 | \( 1 + (6.50 - 2.97i)T + (79.2 - 91.4i)T^{2} \) |
| 13 | \( 1 + (-7.13 + 1.02i)T + (162. - 47.6i)T^{2} \) |
| 17 | \( 1 + (-5.52 + 6.37i)T + (-41.1 - 286. i)T^{2} \) |
| 19 | \( 1 + (-5.13 + 4.45i)T + (51.3 - 357. i)T^{2} \) |
| 29 | \( 1 + (0.932 - 1.07i)T + (-119. - 832. i)T^{2} \) |
| 31 | \( 1 + (6.94 - 2.03i)T + (808. - 519. i)T^{2} \) |
| 37 | \( 1 + (-36.5 - 23.4i)T + (568. + 1.24e3i)T^{2} \) |
| 41 | \( 1 + (39.5 - 25.4i)T + (698. - 1.52e3i)T^{2} \) |
| 43 | \( 1 + (29.5 + 8.68i)T + (1.55e3 + 9.99e2i)T^{2} \) |
| 47 | \( 1 + 9.78iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-11.3 + 78.9i)T + (-2.69e3 - 791. i)T^{2} \) |
| 59 | \( 1 + (-6.75 - 46.9i)T + (-3.33e3 + 980. i)T^{2} \) |
| 61 | \( 1 + (-24.4 - 83.1i)T + (-3.13e3 + 2.01e3i)T^{2} \) |
| 67 | \( 1 + (8.95 - 19.6i)T + (-2.93e3 - 3.39e3i)T^{2} \) |
| 71 | \( 1 + (15.0 - 33.0i)T + (-3.30e3 - 3.80e3i)T^{2} \) |
| 73 | \( 1 + (-52.8 + 45.7i)T + (758. - 5.27e3i)T^{2} \) |
| 79 | \( 1 + (-27.7 + 3.99i)T + (5.98e3 - 1.75e3i)T^{2} \) |
| 83 | \( 1 + (-94.6 - 60.8i)T + (2.86e3 + 6.26e3i)T^{2} \) |
| 89 | \( 1 + (-14.2 + 48.6i)T + (-6.66e3 - 4.28e3i)T^{2} \) |
| 97 | \( 1 + (-53.0 + 34.0i)T + (3.90e3 - 8.55e3i)T^{2} \) |
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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
−11.96285158548097421705864190799, −11.39655112895213963271690425967, −10.38011958185540615402633454075, −9.887470543878848367571433418463, −8.764429265516266818520758410003, −7.73111011879613048953559611762, −6.39771163503625408691313622705, −5.15618979243084784185039410993, −3.73725393463590762510568724748, −2.56002280171086208950986923890,
0.40785906242341698630117590916, 1.52690344340859156434919308199, 3.87391817487178976513907858716, 5.50699792533917745878778471193, 6.58997898094404496911647844260, 7.68747317716862757914745754321, 8.053832686627207800692931832775, 9.408594390100722557455131957974, 10.53229704154415015204663068231, 11.53131068184616431697212467080
