L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (−2.12 + 0.690i)5-s − 0.999i·6-s + (3.89 + 2.24i)7-s + 0.999·8-s + (0.499 + 0.866i)9-s + (1.66 + 1.49i)10-s + 0.412·11-s + (−0.866 + 0.499i)12-s + (1.10 − 1.90i)13-s − 4.49i·14-s + (−2.18 − 0.465i)15-s + (−0.5 − 0.866i)16-s + (−1.63 − 2.82i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.951 + 0.308i)5-s − 0.408i·6-s + (1.47 + 0.849i)7-s + 0.353·8-s + (0.166 + 0.288i)9-s + (0.525 + 0.473i)10-s + 0.124·11-s + (−0.249 + 0.144i)12-s + (0.305 − 0.529i)13-s − 1.20i·14-s + (−0.564 − 0.120i)15-s + (−0.125 − 0.216i)16-s + (−0.395 − 0.685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.511065918\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.511065918\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (2.12 - 0.690i)T \) |
| 37 | \( 1 + (-5.42 - 2.74i)T \) |
good | 7 | \( 1 + (-3.89 - 2.24i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 - 0.412T + 11T^{2} \) |
| 13 | \( 1 + (-1.10 + 1.90i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.63 + 2.82i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.06 - 1.19i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 1.63T + 23T^{2} \) |
| 29 | \( 1 - 9.16iT - 29T^{2} \) |
| 31 | \( 1 - 6.51iT - 31T^{2} \) |
| 41 | \( 1 + (1.68 - 2.91i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 9.32T + 43T^{2} \) |
| 47 | \( 1 - 6.71iT - 47T^{2} \) |
| 53 | \( 1 + (-10.0 + 5.82i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.76 + 2.74i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.46 - 3.73i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.71 - 1.56i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.01 - 1.75i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 6.57iT - 73T^{2} \) |
| 79 | \( 1 + (7.72 + 4.46i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.38 + 1.95i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.49 - 1.44i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−10.00817142155787881502766708121, −8.867944133746822745586877824077, −8.443852199176112661743772630907, −7.80817591864912110637085996804, −6.90580980288080762239355619979, −5.29369257254109334175936029244, −4.63874415838376780919775642404, −3.51169985260463785228083508240, −2.67099386804785233465914626707, −1.39824831185356547008083687326,
0.809664401547555983614031397905, 2.03381642320731032023800809090, 3.98425916440393649132862786975, 4.30064088283493885117453829258, 5.49307191369980488576775324056, 6.75263292094668905231219243624, 7.47624596192486853993305420251, 8.145143413723527124166130263695, 8.505058055067676750481348554950, 9.551951459350400229078725913331