L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.445 − 2.19i)5-s + 4.67i·7-s − 0.999i·8-s + (−0.710 − 2.12i)10-s + 3.96·11-s + (0.698 + 0.403i)13-s + (2.33 + 4.04i)14-s + (−0.5 − 0.866i)16-s + (−4.01 + 2.31i)17-s + (3.01 + 3.15i)19-s + (−1.67 − 1.48i)20-s + (3.43 − 1.98i)22-s + (5.52 + 3.19i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.199 − 0.979i)5-s + 1.76i·7-s − 0.353i·8-s + (−0.224 − 0.670i)10-s + 1.19·11-s + (0.193 + 0.111i)13-s + (0.624 + 1.08i)14-s + (−0.125 − 0.216i)16-s + (−0.974 + 0.562i)17-s + (0.691 + 0.722i)19-s + (−0.374 − 0.331i)20-s + (0.731 − 0.422i)22-s + (1.15 + 0.665i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.172i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.776142727\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.776142727\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.445 + 2.19i)T \) |
| 19 | \( 1 + (-3.01 - 3.15i)T \) |
good | 7 | \( 1 - 4.67iT - 7T^{2} \) |
| 11 | \( 1 - 3.96T + 11T^{2} \) |
| 13 | \( 1 + (-0.698 - 0.403i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (4.01 - 2.31i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-5.52 - 3.19i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.03 + 3.51i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.35T + 31T^{2} \) |
| 37 | \( 1 + 2.19iT - 37T^{2} \) |
| 41 | \( 1 + (-2.02 - 3.51i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.36 - 3.67i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-8.12 - 4.69i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.34 - 0.778i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.94 + 6.83i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.20 + 3.81i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.42 - 3.70i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.08 + 1.87i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-10.2 + 5.91i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.67 + 8.09i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9.92iT - 83T^{2} \) |
| 89 | \( 1 + (-9.13 + 15.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.7 - 6.20i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.189661422881131437239373063055, −8.822494309583103039364803702700, −7.909512111929420559375857949034, −6.47221714409595867491165496168, −5.99358414375544758966164299585, −5.17610805278716098434932253215, −4.42540232257045702895772961702, −3.36345101809575577338788740519, −2.20707703192014681131640340920, −1.32139572660904215133971814284,
1.00812927847500369986293703028, 2.62820841457768282751685462323, 3.59690757756722777908419361623, 4.25343985550753296512477836483, 5.17529180591934832425459570283, 6.50968896916676517397293879072, 6.93150393347421770262482036557, 7.24572871466425497144517718226, 8.467963387947485839970336713407, 9.395677243575917898597910740073