L(s) = 1 | + (−0.766 + 0.642i)2-s + (−0.476 − 0.173i)3-s + (0.173 − 0.984i)4-s + (0.476 − 0.173i)6-s + (0.753 − 1.30i)7-s + (0.500 + 0.866i)8-s + (−2.10 − 1.76i)9-s + (1.03 + 1.79i)11-s + (−0.253 + 0.439i)12-s + (−5.26 + 1.91i)13-s + (0.261 + 1.48i)14-s + (−0.939 − 0.342i)16-s + (−2.67 + 2.24i)17-s + 2.74·18-s + (3.36 − 2.77i)19-s + ⋯ |
L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.275 − 0.100i)3-s + (0.0868 − 0.492i)4-s + (0.194 − 0.0708i)6-s + (0.284 − 0.493i)7-s + (0.176 + 0.306i)8-s + (−0.700 − 0.587i)9-s + (0.312 + 0.540i)11-s + (−0.0732 + 0.126i)12-s + (−1.45 + 0.531i)13-s + (0.0699 + 0.396i)14-s + (−0.234 − 0.0855i)16-s + (−0.648 + 0.544i)17-s + 0.646·18-s + (0.771 − 0.636i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.303114 + 0.510881i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.303114 + 0.510881i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.36 + 2.77i)T \) |
good | 3 | \( 1 + (0.476 + 0.173i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (-0.753 + 1.30i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.03 - 1.79i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (5.26 - 1.91i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (2.67 - 2.24i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (1.20 - 6.83i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-6.38 - 5.35i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.844 + 1.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.297T + 37T^{2} \) |
| 41 | \( 1 + (2.94 + 1.07i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.40 - 7.95i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.39 - 2.84i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.25 + 7.10i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (8.89 - 7.46i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (2.23 - 12.6i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (7.34 + 6.16i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.508 - 2.88i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-5.58 - 2.03i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (11.2 + 4.08i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.94 - 3.37i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (5.44 - 1.98i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (12.3 - 10.3i)T + (16.8 - 95.5i)T^{2} \) |
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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
−10.12712601289688822412925082627, −9.406848824484770225230265064017, −8.763974123308111732611773054469, −7.58277310721576133802657936022, −7.08276385238266121655750303410, −6.21125783133435134673700103997, −5.16238242920406139998401104967, −4.31502301317510340956388845886, −2.81005663343785369017360490616, −1.33654275385900114967288119090,
0.35610788078987760720981246297, 2.23779390876423324948021544859, 2.97010960869478114134206355467, 4.51086449871327109574527860025, 5.31965302691621882300281634175, 6.32983605393929284442277006288, 7.45393128450026709041605727662, 8.274048026348812659728753850840, 8.878601740253581710690307235264, 9.932360287857174599120319044898