L(s) = 1 | + (−2.18 + 1.58i)2-s + (1.68 − 1.22i)3-s + (1.63 − 5.04i)4-s + (−0.994 − 3.06i)5-s + (−1.73 + 5.34i)6-s + (0.649 − 0.472i)7-s + (2.75 + 8.49i)8-s + (0.410 − 1.26i)9-s + (7.03 + 5.11i)10-s + (−1.76 + 1.27i)11-s + (−3.41 − 10.4i)12-s + (2.15 + 1.56i)13-s + (−0.670 + 2.06i)14-s + (−5.41 − 3.93i)15-s + (−10.9 − 7.94i)16-s + (1.63 + 1.18i)17-s + ⋯ |
L(s) = 1 | + (−1.54 + 1.12i)2-s + (0.971 − 0.706i)3-s + (0.819 − 2.52i)4-s + (−0.444 − 1.36i)5-s + (−0.709 + 2.18i)6-s + (0.245 − 0.178i)7-s + (0.975 + 3.00i)8-s + (0.136 − 0.421i)9-s + (2.22 + 1.61i)10-s + (−0.531 + 0.385i)11-s + (−0.984 − 3.02i)12-s + (0.598 + 0.435i)13-s + (−0.179 + 0.551i)14-s + (−1.39 − 1.01i)15-s + (−2.73 − 1.98i)16-s + (0.396 + 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.585838 - 0.0583859i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.585838 - 0.0583859i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 71 | \( 1 + (-5.71 - 6.19i)T \) |
good | 2 | \( 1 + (2.18 - 1.58i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-1.68 + 1.22i)T + (0.927 - 2.85i)T^{2} \) |
| 5 | \( 1 + (0.994 + 3.06i)T + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (-0.649 + 0.472i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (1.76 - 1.27i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-2.15 - 1.56i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.63 - 1.18i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.21 - 0.881i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 6.57T + 23T^{2} \) |
| 29 | \( 1 + (0.0382 + 0.117i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.464 - 0.337i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + 5.83T + 37T^{2} \) |
| 41 | \( 1 + 8.59T + 41T^{2} \) |
| 43 | \( 1 + (-0.557 - 1.71i)T + (-34.7 + 25.2i)T^{2} \) |
| 47 | \( 1 + (-8.39 - 6.09i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (4.03 + 12.4i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.69 + 11.3i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.35 - 0.981i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (0.173 - 0.535i)T + (-54.2 - 39.3i)T^{2} \) |
| 73 | \( 1 + (1.21 - 0.884i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.21 + 6.81i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.61 - 11.1i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.52 - 4.69i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.96665926704508049801505282148, −13.86512326820464480449117596875, −12.65901672571083566657251741708, −10.93637502891101082222291953681, −9.432687275123693049638160376291, −8.510736073779759665929404219902, −8.033932831173493298030159566756, −6.93510093975587795625800909165, −5.14035805562653899261156169725, −1.47718653552201638364070714439,
2.77049752772748932450342951887, 3.51717706177472345292274758075, 7.13256483514187258440730088721, 8.249459292170318397361368348086, 9.091736003449581012332661479482, 10.37846253393884593774311243386, 10.83520852540032991549175309401, 11.96683997160361439639792443114, 13.57094203074318149436250068236, 15.03913699544302363585771580008