L(s) = 1 | + (−1.36 + 0.381i)2-s + (1.60 − 0.430i)3-s + (1.70 − 1.03i)4-s + (−1.94 − 0.522i)5-s + (−2.02 + 1.20i)6-s + (−1.93 + 2.06i)8-s + (−0.196 + 0.113i)9-s + (2.85 − 0.0319i)10-s + (1.63 + 6.09i)11-s + (2.30 − 2.40i)12-s + (−1.13 − 1.13i)13-s − 3.35·15-s + (1.84 − 3.55i)16-s + (0.960 − 1.66i)17-s + (0.224 − 0.229i)18-s + (−1.63 + 6.09i)19-s + ⋯ |
L(s) = 1 | + (−0.962 + 0.269i)2-s + (0.928 − 0.248i)3-s + (0.854 − 0.519i)4-s + (−0.871 − 0.233i)5-s + (−0.827 + 0.490i)6-s + (−0.682 + 0.730i)8-s + (−0.0655 + 0.0378i)9-s + (0.902 − 0.0101i)10-s + (0.492 + 1.83i)11-s + (0.664 − 0.694i)12-s + (−0.314 − 0.314i)13-s − 0.867·15-s + (0.460 − 0.887i)16-s + (0.233 − 0.403i)17-s + (0.0529 − 0.0541i)18-s + (−0.374 + 1.39i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0406 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0406 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.602142 + 0.627148i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.602142 + 0.627148i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.36 - 0.381i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.60 + 0.430i)T + (2.59 - 1.5i)T^{2} \) |
| 5 | \( 1 + (1.94 + 0.522i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-1.63 - 6.09i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (1.13 + 1.13i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.960 + 1.66i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.63 - 6.09i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (0.924 - 0.533i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.08 - 5.08i)T + 29iT^{2} \) |
| 31 | \( 1 + (-0.198 + 0.343i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.327 - 0.0877i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 7.26iT - 41T^{2} \) |
| 43 | \( 1 + (-1.75 + 1.75i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.08 - 1.87i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.111 - 0.415i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.32 - 8.67i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.72 + 10.1i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (-7.87 + 2.10i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 2.78iT - 71T^{2} \) |
| 73 | \( 1 + (2.05 + 1.18i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.10 - 8.84i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.17 + 1.17i)T + 83iT^{2} \) |
| 89 | \( 1 + (11.0 - 6.37i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2.21T + 97T^{2} \) |
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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
−10.12097037120915617455893283908, −9.608152971555310225319590975425, −8.636845378672523048455599220104, −7.939229099074448440824851759740, −7.47436011071631335842549291495, −6.57721497845852993837403279991, −5.19808890951642126705464783604, −3.97717828814882504891603686642, −2.66653873520303779690465086189, −1.57550743003390362004507409397,
0.53140767123236082285732081179, 2.47156771810868955261855314868, 3.33917825869782955287660474296, 4.07058662742957447242268807526, 5.94272782246465609301506829630, 6.87399151367734929234507874828, 7.897202990538152546321380308474, 8.531996834350426355515457236660, 8.975764534138145532578501853029, 9.914828995081989023362882504482