| L(s) = 1 | + (−0.454 − 0.381i)2-s + (1.81 − 0.660i)3-s + (−0.286 − 1.62i)4-s + (−0.173 + 0.984i)5-s + (−1.07 − 0.392i)6-s + (−0.530 − 0.918i)7-s + (−1.08 + 1.87i)8-s + (0.555 − 0.466i)9-s + (0.454 − 0.381i)10-s + (−0.0983 + 0.170i)11-s + (−1.58 − 2.75i)12-s + (4.96 + 1.80i)13-s + (−0.109 + 0.620i)14-s + (0.335 + 1.90i)15-s + (−1.88 + 0.686i)16-s + (0.540 + 0.453i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 − 0.269i)2-s + (1.04 − 0.381i)3-s + (−0.143 − 0.811i)4-s + (−0.0776 + 0.440i)5-s + (−0.439 − 0.160i)6-s + (−0.200 − 0.347i)7-s + (−0.382 + 0.663i)8-s + (0.185 − 0.155i)9-s + (0.143 − 0.120i)10-s + (−0.0296 + 0.0513i)11-s + (−0.458 − 0.794i)12-s + (1.37 + 0.501i)13-s + (−0.0292 + 0.165i)14-s + (0.0865 + 0.490i)15-s + (−0.471 + 0.171i)16-s + (0.130 + 0.109i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.953582 - 0.466884i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.953582 - 0.466884i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 + (4.24 + 0.983i)T \) |
| good | 2 | \( 1 + (0.454 + 0.381i)T + (0.347 + 1.96i)T^{2} \) |
| 3 | \( 1 + (-1.81 + 0.660i)T + (2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (0.530 + 0.918i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.0983 - 0.170i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.96 - 1.80i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.540 - 0.453i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.15 - 6.52i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.59 - 2.17i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.95 + 6.85i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.09T + 37T^{2} \) |
| 41 | \( 1 + (-1.27 + 0.463i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.56 + 8.87i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.69 + 3.09i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.924 + 5.24i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (8.41 + 7.05i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.04 - 11.5i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (2.34 - 1.96i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.434 + 2.46i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (6.15 - 2.24i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-11.5 + 4.19i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-2.01 - 3.49i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (16.4 + 5.99i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (1.02 + 0.861i)T + (16.8 + 95.5i)T^{2} \) |
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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
−13.79952861410700024881185238816, −13.22980242579043055281470642298, −11.40478551292271112214494956463, −10.61063197072456328128400557812, −9.329652448014510703960425969367, −8.536092598131456556983724291502, −7.18412554671242274541798229661, −5.80760517315901081263838067498, −3.70365300966170389146581462920, −1.94753655248776577795791107246,
2.99720492964782134570892036342, 4.17227855107211888746771007403, 6.23522124045540974715672922481, 7.909933543149059132879575200152, 8.660891763240449719487390137131, 9.256703165708221187698820950407, 10.79279156110545962389634396497, 12.34382972435844517963117958099, 13.07873428056999828246946531558, 14.18928494918356173724749400195
