L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (−1.66 − 1.49i)5-s + 1.00·6-s + (−0.170 + 0.170i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (0.120 + 2.23i)10-s + 3.43·11-s + (−0.707 − 0.707i)12-s + (−4.54 + 4.54i)13-s + 0.240·14-s + (2.23 − 0.120i)15-s − 1.00·16-s + (0.537 − 0.537i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (−0.744 − 0.668i)5-s + 0.408·6-s + (−0.0643 + 0.0643i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.0380 + 0.706i)10-s + 1.03·11-s + (−0.204 − 0.204i)12-s + (−1.25 + 1.25i)13-s + 0.0643·14-s + (0.576 − 0.0310i)15-s − 0.250·16-s + (0.130 − 0.130i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.639 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.606780 + 0.284339i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.606780 + 0.284339i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 + 0.707i)T \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.66 + 1.49i)T \) |
| 19 | \( 1 + (2.42 + 3.62i)T \) |
good | 7 | \( 1 + (0.170 - 0.170i)T - 7iT^{2} \) |
| 11 | \( 1 - 3.43T + 11T^{2} \) |
| 13 | \( 1 + (4.54 - 4.54i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.537 + 0.537i)T - 17iT^{2} \) |
| 23 | \( 1 + (-5.15 - 5.15i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.37T + 29T^{2} \) |
| 31 | \( 1 - 5.37iT - 31T^{2} \) |
| 37 | \( 1 + (-5.54 - 5.54i)T + 37iT^{2} \) |
| 41 | \( 1 - 3.68iT - 41T^{2} \) |
| 43 | \( 1 + (-2.29 - 2.29i)T + 43iT^{2} \) |
| 47 | \( 1 + (9.57 - 9.57i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.93 + 1.93i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 - 1.65T + 61T^{2} \) |
| 67 | \( 1 + (0.481 + 0.481i)T + 67iT^{2} \) |
| 71 | \( 1 + 8.93iT - 71T^{2} \) |
| 73 | \( 1 + (-8.74 - 8.74i)T + 73iT^{2} \) |
| 79 | \( 1 + 1.15T + 79T^{2} \) |
| 83 | \( 1 + (9.97 + 9.97i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.10T + 89T^{2} \) |
| 97 | \( 1 + (10.8 + 10.8i)T + 97iT^{2} \) |
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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
−11.14704965925167045427741292139, −9.726615168759100346534046909593, −9.315230794876836351766134441970, −8.496242145063442920181786265455, −7.30940398973003148615118325243, −6.54033796640336863604828543261, −4.84730394200402715724007968122, −4.37501153721245148033159384194, −3.04500388466567090020514423820, −1.26132246646093364993942878734,
0.54232341356013299088050799064, 2.51145575258180944356449347853, 3.98610759283951976236747396894, 5.23383671433754876663525117903, 6.37551022822736508890745829134, 7.02508403190380703760400856156, 7.84504263484200542494467317526, 8.592693024052786932364199151369, 9.906429388777959324951676853261, 10.50453623174561238426005202134