L(s) = 1 | + (−2.59 + 1.5i)3-s + (−2.73 + 1.58i)5-s − 3.16·7-s + (3 − 5.19i)9-s + 3i·11-s + (−5.47 − 3.16i)13-s + (4.74 − 8.21i)15-s + (−2 − 3.46i)17-s + (−2.59 − 3.5i)19-s + (8.21 − 4.74i)21-s + (−4.74 + 8.21i)23-s + (2.5 − 4.33i)25-s + 9i·27-s + (2.73 + 1.58i)29-s − 3.16·31-s + ⋯ |
L(s) = 1 | + (−1.49 + 0.866i)3-s + (−1.22 + 0.707i)5-s − 1.19·7-s + (1 − 1.73i)9-s + 0.904i·11-s + (−1.51 − 0.877i)13-s + (1.22 − 2.12i)15-s + (−0.485 − 0.840i)17-s + (−0.596 − 0.802i)19-s + (1.79 − 1.03i)21-s + (−0.989 + 1.71i)23-s + (0.5 − 0.866i)25-s + 1.73i·27-s + (0.508 + 0.293i)29-s − 0.567·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.09222143456\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09222143456\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.59 + 3.5i)T \) |
good | 3 | \( 1 + (2.59 - 1.5i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2.73 - 1.58i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 - 3iT - 11T^{2} \) |
| 13 | \( 1 + (5.47 + 3.16i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (4.74 - 8.21i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.73 - 1.58i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.16T + 31T^{2} \) |
| 37 | \( 1 - 3.16iT - 37T^{2} \) |
| 41 | \( 1 + (-1.5 - 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (8.66 - 5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.58 - 2.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.06 + 3.5i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.73 - 1.58i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.33 - 2.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.16 + 5.47i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 7iT - 83T^{2} \) |
| 89 | \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.5 - 7.79i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.842059213690882464767748749284, −9.488068423187399273572096884097, −7.83563527398309296722093874391, −7.01645217944862882548714563863, −6.57501756675565271496926103676, −5.35921274770510768020411900301, −4.67609688321778936454490537540, −3.77430359607055028566675913111, −2.82659163759581521881084738684, −0.13203095838916446294954479009,
0.39176378277133973314657138967, 2.12591733517300823617626605424, 3.84804938759556932956895579481, 4.59247431138898073974937916069, 5.65529738258171969722423299648, 6.49794322087648307186246261039, 6.95130561523116885571904014024, 8.011476836480493765857469815835, 8.660285327155052263623109547370, 9.933822256712994102873477338436