L(s) = 1 | − i·2-s + (0.866 − 0.5i)3-s − 4-s + (−1.23 − 1.86i)5-s + (−0.5 − 0.866i)6-s + (1.73 − i)7-s + i·8-s + (0.499 − 0.866i)9-s + (−1.86 + 1.23i)10-s + (2.5 − 4.33i)11-s + (−0.866 + 0.5i)12-s + (−1.73 − i)13-s + (−1 − 1.73i)14-s + (−2 − i)15-s + 16-s + (−2.59 + 1.5i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.499 − 0.288i)3-s − 0.5·4-s + (−0.550 − 0.834i)5-s + (−0.204 − 0.353i)6-s + (0.654 − 0.377i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.590 + 0.389i)10-s + (0.753 − 1.30i)11-s + (−0.249 + 0.144i)12-s + (−0.480 − 0.277i)13-s + (−0.267 − 0.462i)14-s + (−0.516 − 0.258i)15-s + 0.250·16-s + (−0.630 + 0.363i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.953 + 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.232237 - 1.51155i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.232237 - 1.51155i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (1.23 + 1.86i)T \) |
| 31 | \( 1 + (2 + 5.19i)T \) |
good | 7 | \( 1 + (-1.73 + i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.73 + i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.59 - 1.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 37 | \( 1 + (4.33 - 2.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9iT - 47T^{2} \) |
| 53 | \( 1 + (6.92 + 4i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + (6.06 + 3.5i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.19 - 3i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.19 - 3i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−9.613615173222400871179033202433, −8.707509370507257370572669129636, −8.291459177683712603038891278421, −7.47295482594320977792938985169, −6.17065328445446170290452624816, −5.02913024755298323214525317011, −4.08357345122381079003775041300, −3.32587196482208242300162710082, −1.82955640197541103442578396430, −0.70305337668896462908401572699,
1.98645003579625932602866018188, 3.24477575697928932429719072066, 4.42521739603077710208987768990, 4.97313790205768912551746053087, 6.43316159839270667801433492934, 7.22536084614438592387218736465, 7.67072362080928654624130621883, 8.826709261524974662182995306888, 9.382560098260601665565034080398, 10.27599056833427539318206561480