L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (1.51 + 1.64i)5-s − 0.999i·6-s + (0.0701 − 0.0404i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (−2.18 + 0.483i)10-s − 3.93·11-s + (0.866 + 0.499i)12-s + (1.78 + 3.09i)13-s + 0.0809i·14-s + (−2.13 − 0.672i)15-s + (−0.5 + 0.866i)16-s + (0.563 − 0.975i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.675 + 0.737i)5-s − 0.408i·6-s + (0.0264 − 0.0152i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.690 + 0.152i)10-s − 1.18·11-s + (0.249 + 0.144i)12-s + (0.495 + 0.857i)13-s + 0.0216i·14-s + (−0.550 − 0.173i)15-s + (−0.125 + 0.216i)16-s + (0.136 − 0.236i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6589530661\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6589530661\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.51 - 1.64i)T \) |
| 37 | \( 1 + (-1.49 - 5.89i)T \) |
good | 7 | \( 1 + (-0.0701 + 0.0404i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 3.93T + 11T^{2} \) |
| 13 | \( 1 + (-1.78 - 3.09i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.563 + 0.975i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.70 - 0.983i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 2.93T + 23T^{2} \) |
| 29 | \( 1 + 2.65iT - 29T^{2} \) |
| 31 | \( 1 - 6.08iT - 31T^{2} \) |
| 41 | \( 1 + (3.72 + 6.44i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 5.69T + 43T^{2} \) |
| 47 | \( 1 - 10.7iT - 47T^{2} \) |
| 53 | \( 1 + (0.118 + 0.0685i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.46 + 1.42i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.34 + 4.81i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.06 - 1.76i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.49 + 6.04i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 5.98iT - 73T^{2} \) |
| 79 | \( 1 + (3.95 - 2.28i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (10.6 + 6.14i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (11.3 + 6.52i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.882T + 97T^{2} \) |
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
−10.20568823345775622920905550290, −9.598010261680794825881061629196, −8.627242985855074781512459657533, −7.72519265489444410130463716533, −6.79655778090771193649082586923, −6.16082528544067061140075002525, −5.37633129084395739368676953208, −4.43981983333196131634250692862, −3.07447450750449067783727586892, −1.72508533571879713419748778384,
0.33988567549595269798953887570, 1.68283845130453047618243333713, 2.73223301541423961961796571063, 4.12218649027838157703902461465, 5.26636179046074079881651969375, 5.75115093676467175977884605239, 6.93885013752410359233181765952, 8.149747026043515876575859075978, 8.415612873515036644865570631348, 9.669240386782114820316847371745