| L(s) = 1 | + (−0.654 − 0.755i)2-s + (−0.578 − 0.371i)3-s + (−0.142 + 0.989i)4-s + (−0.415 + 0.909i)5-s + (0.0978 + 0.680i)6-s + (−2.88 − 0.847i)7-s + (0.841 − 0.540i)8-s + (−1.04 − 2.29i)9-s + (0.959 − 0.281i)10-s + (−2.12 + 2.44i)11-s + (0.450 − 0.519i)12-s + (−4.24 + 1.24i)13-s + (1.24 + 2.73i)14-s + (0.578 − 0.371i)15-s + (−0.959 − 0.281i)16-s + (−1.00 − 6.98i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (−0.334 − 0.214i)3-s + (−0.0711 + 0.494i)4-s + (−0.185 + 0.406i)5-s + (0.0399 + 0.277i)6-s + (−1.09 − 0.320i)7-s + (0.297 − 0.191i)8-s + (−0.349 − 0.766i)9-s + (0.303 − 0.0890i)10-s + (−0.639 + 0.738i)11-s + (0.130 − 0.150i)12-s + (−1.17 + 0.345i)13-s + (0.333 + 0.731i)14-s + (0.149 − 0.0960i)15-s + (−0.239 − 0.0704i)16-s + (−0.243 − 1.69i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0171206 + 0.0774842i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0171206 + 0.0774842i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 5 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (0.769 - 4.73i)T \) |
| good | 3 | \( 1 + (0.578 + 0.371i)T + (1.24 + 2.72i)T^{2} \) |
| 7 | \( 1 + (2.88 + 0.847i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (2.12 - 2.44i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (4.24 - 1.24i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (1.00 + 6.98i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.514 - 3.57i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.568 - 3.95i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-1.67 + 1.07i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (4.17 + 9.13i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-3.00 + 6.57i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (9.32 + 5.99i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 6.41T + 47T^{2} \) |
| 53 | \( 1 + (-12.6 - 3.72i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-7.33 + 2.15i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (5.88 - 3.78i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (2.10 + 2.43i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (5.33 + 6.15i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.990 - 6.88i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (4.21 - 1.23i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-1.85 - 4.05i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (4.28 + 2.75i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-1.67 + 3.66i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.91923594687347270942988401217, −10.56955960464524556120699550878, −9.783675013551820179520633010397, −9.072242479265351937030987700597, −7.32233969964387912254560695359, −6.99839092102986628854613107523, −5.40677387128919623129271661124, −3.75829300933709222904460362159, −2.53847985563236705184142747759, −0.07250741400641113318635805799,
2.71095266711457330056649473662, 4.62016358322131445331249597649, 5.69231128381756049165637317685, 6.60542057879520048164341076576, 8.033878734354513990539880775662, 8.639109037662329790205508647911, 9.968241879571297449003150444632, 10.52016694981961652783090160653, 11.72918515257048102792980993763, 12.87413147347843909695696296201
