| L(s) = 1 | + (0.744 + 0.373i)2-s + (0.703 + 1.58i)3-s + (−0.779 − 1.04i)4-s + (0.345 + 1.15i)5-s + (−0.0676 + 1.44i)6-s + (−0.520 − 1.20i)7-s + (−0.478 − 2.71i)8-s + (−2.00 + 2.22i)9-s + (−0.174 + 0.989i)10-s + (2.11 − 0.501i)11-s + (1.10 − 1.97i)12-s + (−3.80 − 2.50i)13-s + (0.0636 − 1.09i)14-s + (−1.58 + 1.36i)15-s + (−0.0912 + 0.304i)16-s + (3.54 − 1.28i)17-s + ⋯ |
| L(s) = 1 | + (0.526 + 0.264i)2-s + (0.406 + 0.913i)3-s + (−0.389 − 0.523i)4-s + (0.154 + 0.516i)5-s + (−0.0276 + 0.588i)6-s + (−0.196 − 0.456i)7-s + (−0.169 − 0.958i)8-s + (−0.669 + 0.742i)9-s + (−0.0551 + 0.312i)10-s + (0.637 − 0.151i)11-s + (0.320 − 0.569i)12-s + (−1.05 − 0.694i)13-s + (0.0170 − 0.292i)14-s + (−0.409 + 0.351i)15-s + (−0.0228 + 0.0761i)16-s + (0.859 − 0.312i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.13740 + 0.418027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.13740 + 0.418027i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.703 - 1.58i)T \) |
| good | 2 | \( 1 + (-0.744 - 0.373i)T + (1.19 + 1.60i)T^{2} \) |
| 5 | \( 1 + (-0.345 - 1.15i)T + (-4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (0.520 + 1.20i)T + (-4.80 + 5.09i)T^{2} \) |
| 11 | \( 1 + (-2.11 + 0.501i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (3.80 + 2.50i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (-3.54 + 1.28i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (2.50 + 0.911i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (2.38 - 5.53i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (-0.241 - 4.15i)T + (-28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (7.40 - 0.865i)T + (30.1 - 7.14i)T^{2} \) |
| 37 | \( 1 + (-7.47 - 6.27i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-5.17 + 2.59i)T + (24.4 - 32.8i)T^{2} \) |
| 43 | \( 1 + (-3.46 - 3.67i)T + (-2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-2.97 - 0.348i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (6.22 - 10.7i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (10.6 + 2.51i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (-7.04 + 9.45i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (-0.0482 + 0.828i)T + (-66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (-1.25 + 7.14i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.41 + 7.99i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (10.2 + 5.16i)T + (47.1 + 63.3i)T^{2} \) |
| 83 | \( 1 + (4.26 + 2.14i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (-0.578 - 3.28i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (0.390 - 1.30i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−14.49798531887954687776046986410, −13.88519213860512241837981232303, −12.59344630341229735809923326573, −10.91127604426353929450572479093, −9.996849513610254956064548183517, −9.225182394302638019747765128797, −7.48250397780019784360537381140, −5.91812524041052484590700227905, −4.67442667456820543059632374447, −3.31757211898587386784856436161,
2.39507635433655029372911158548, 4.16314192954395286162154840234, 5.82944469430959558126534426016, 7.39741712352485313953142164376, 8.600371352579214778816855464459, 9.449623837623398221116256944433, 11.55078706773173010556821584477, 12.59305741633429464623828691485, 12.73991577883053082202673608077, 14.28152636617441902296481287398
