L(s) = 1 | + (0.866 + 0.5i)2-s + (−1.28 + 1.16i)3-s + (0.499 + 0.866i)4-s − 5-s + (−1.69 + 0.361i)6-s + (−0.555 − 2.58i)7-s + 0.999i·8-s + (0.307 − 2.98i)9-s + (−0.866 − 0.5i)10-s − 4.07i·11-s + (−1.64 − 0.533i)12-s + (−5.20 − 3.00i)13-s + (0.812 − 2.51i)14-s + (1.28 − 1.16i)15-s + (−0.5 + 0.866i)16-s + (−0.641 + 1.11i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.742 + 0.669i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (−0.691 + 0.147i)6-s + (−0.209 − 0.977i)7-s + 0.353i·8-s + (0.102 − 0.994i)9-s + (−0.273 − 0.158i)10-s − 1.22i·11-s + (−0.475 − 0.154i)12-s + (−1.44 − 0.832i)13-s + (0.217 − 0.672i)14-s + (0.332 − 0.299i)15-s + (−0.125 + 0.216i)16-s + (−0.155 + 0.269i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.749407 - 0.491330i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.749407 - 0.491330i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.28 - 1.16i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (0.555 + 2.58i)T \) |
good | 11 | \( 1 + 4.07iT - 11T^{2} \) |
| 13 | \( 1 + (5.20 + 3.00i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.641 - 1.11i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.90 + 1.67i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.18iT - 23T^{2} \) |
| 29 | \( 1 + (-6.21 + 3.59i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (6.79 - 3.92i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.90 + 5.02i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.36 + 2.37i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.43 + 4.22i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.13 + 7.16i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (7.48 + 4.32i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (0.630 + 1.09i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.56 + 1.48i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.31 - 7.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.1iT - 71T^{2} \) |
| 73 | \( 1 + (6.02 + 3.47i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.09 + 10.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.87 - 11.9i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.29 + 12.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (16.8 - 9.74i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.58457137180192881173801034711, −9.801755968662116440963268495521, −8.624472350398983300873177493194, −7.50377005714802818214323346587, −6.82977208650504301494512955015, −5.66273527805387917472218990619, −4.97696281364427852693587530036, −3.89965331093658811032033543278, −3.12950281233084874325303038163, −0.42596909628755233020467051758,
1.80316904556372218440248277197, 2.79921769398120015236735029410, 4.57768519287402911526685693228, 5.05540042410932620847340304550, 6.25275983427477600878824767112, 7.05395408076513446067247247291, 7.81088729019463929887159141341, 9.249853622161305973674672522448, 9.980316472792767623587930171219, 11.07559420243131439794482190874