| L(s) = 1 | + (0.654 − 0.755i)2-s + (2.24 − 1.44i)3-s + (−0.142 − 0.989i)4-s + (1.60 + 3.51i)5-s + (0.380 − 2.64i)6-s + (−0.959 + 0.281i)7-s + (−0.841 − 0.540i)8-s + (1.72 − 3.77i)9-s + (3.70 + 1.08i)10-s + (−3.45 − 3.98i)11-s + (−1.75 − 2.02i)12-s + (1.18 + 0.347i)13-s + (−0.415 + 0.909i)14-s + (8.69 + 5.58i)15-s + (−0.959 + 0.281i)16-s + (−0.676 + 4.70i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (1.29 − 0.834i)3-s + (−0.0711 − 0.494i)4-s + (0.717 + 1.57i)5-s + (0.155 − 1.08i)6-s + (−0.362 + 0.106i)7-s + (−0.297 − 0.191i)8-s + (0.575 − 1.25i)9-s + (1.17 + 0.344i)10-s + (−1.04 − 1.20i)11-s + (−0.505 − 0.583i)12-s + (0.328 + 0.0963i)13-s + (−0.111 + 0.243i)14-s + (2.24 + 1.44i)15-s + (−0.239 + 0.0704i)16-s + (−0.163 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.18921 - 1.08603i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.18921 - 1.08603i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (0.959 - 0.281i)T \) |
| 23 | \( 1 + (1.10 + 4.66i)T \) |
| good | 3 | \( 1 + (-2.24 + 1.44i)T + (1.24 - 2.72i)T^{2} \) |
| 5 | \( 1 + (-1.60 - 3.51i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (3.45 + 3.98i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.18 - 0.347i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (0.676 - 4.70i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.293 - 2.04i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-0.814 + 5.66i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (6.14 + 3.94i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (0.701 - 1.53i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-3.34 - 7.32i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (5.09 - 3.27i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 3.43T + 47T^{2} \) |
| 53 | \( 1 + (-5.43 + 1.59i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (6.45 + 1.89i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-11.2 - 7.20i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (3.32 - 3.84i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-4.48 + 5.17i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.0550 - 0.383i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-11.4 - 3.35i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-3.63 + 7.96i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-4.61 + 2.96i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (5.29 + 11.5i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.38761241641973201322563175413, −10.56182875137528963502306401493, −9.825459721190255398183148290174, −8.568947452716173170128311250197, −7.73225436965559613242553204010, −6.47753937205950722470137802518, −5.90904248973874175479146331614, −3.63387346726503003236653907428, −2.85113974231318966777069888499, −2.05424076360252533030642353181,
2.22589605448085910996655236474, 3.64142706357472542624023226633, 4.88210057662458457934462039883, 5.29790834503906030969086326580, 7.13652887595348134917313624256, 8.125309260774064052154100049800, 9.142886220765066114709097365124, 9.410014344006989326918091275010, 10.45959851165002673819039944044, 12.19778622018912035528458255940
