| L(s) = 1 | + (−1.02 + 0.950i)2-s + (−1.96 + 0.295i)3-s + (−0.00340 + 0.0454i)4-s + (−1.11 + 2.83i)5-s + (1.72 − 2.16i)6-s + (2.44 − 1.01i)7-s + (−1.78 − 2.23i)8-s + (0.892 − 0.275i)9-s + (−1.55 − 3.96i)10-s + (4.91 + 1.51i)11-s + (−0.00676 − 0.0902i)12-s + (0.369 + 1.61i)13-s + (−1.54 + 3.36i)14-s + (1.34 − 5.88i)15-s + (3.86 + 0.582i)16-s + (−2.42 + 1.65i)17-s + ⋯ |
| L(s) = 1 | + (−0.724 + 0.672i)2-s + (−1.13 + 0.170i)3-s + (−0.00170 + 0.0227i)4-s + (−0.497 + 1.26i)5-s + (0.705 − 0.885i)6-s + (0.923 − 0.382i)7-s + (−0.630 − 0.790i)8-s + (0.297 − 0.0917i)9-s + (−0.491 − 1.25i)10-s + (1.48 + 0.457i)11-s + (−0.00195 − 0.0260i)12-s + (0.102 + 0.448i)13-s + (−0.412 + 0.898i)14-s + (0.347 − 1.52i)15-s + (0.965 + 0.145i)16-s + (−0.587 + 0.400i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.568 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.568 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.194060 + 0.369720i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.194060 + 0.369720i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.44 + 1.01i)T \) |
| good | 2 | \( 1 + (1.02 - 0.950i)T + (0.149 - 1.99i)T^{2} \) |
| 3 | \( 1 + (1.96 - 0.295i)T + (2.86 - 0.884i)T^{2} \) |
| 5 | \( 1 + (1.11 - 2.83i)T + (-3.66 - 3.40i)T^{2} \) |
| 11 | \( 1 + (-4.91 - 1.51i)T + (9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.369 - 1.61i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (2.42 - 1.65i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (0.170 + 0.295i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.94 + 2.01i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-5.41 - 2.60i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (-1.30 + 2.26i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.120 + 1.60i)T + (-36.5 + 5.51i)T^{2} \) |
| 41 | \( 1 + (2.03 + 2.54i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (2.88 - 3.61i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-7.48 + 6.94i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.217 + 2.89i)T + (-52.4 - 7.89i)T^{2} \) |
| 59 | \( 1 + (3.40 + 8.66i)T + (-43.2 + 40.1i)T^{2} \) |
| 61 | \( 1 + (-0.385 - 5.14i)T + (-60.3 + 9.09i)T^{2} \) |
| 67 | \( 1 + (5.99 - 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.67 - 0.805i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-9.61 - 8.92i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (6.91 + 11.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.194 + 0.852i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (0.454 - 0.140i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 7.70T + 97T^{2} \) |
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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
−16.32337235855666525206690495568, −15.11318938887310210854575770546, −14.22215110556126687575225678567, −12.05893161256893778825084269475, −11.35571013145049206249427381236, −10.26799462540456704564800096979, −8.574977079282891286257527623766, −7.11363485146023512263379196963, −6.39469845847989057103641579133, −4.14541226270001693919793854843,
1.08918189451015147236216089966, 4.78249789899734697249822392503, 6.03139271967563252570653889461, 8.328454080612740158560101028664, 9.162159259083217567465632933826, 10.83055245771222013192554328442, 11.84623713557999503002751403500, 12.09012584517780251451066001795, 14.06206188062507951775152126451, 15.52182297507189821073116041833
