| L(s) = 1 | + (1.39 − 1.16i)2-s + (−0.166 − 0.0605i)3-s + (0.226 − 1.28i)4-s + (−0.173 − 0.984i)5-s + (−0.302 + 0.110i)6-s + (−0.536 + 0.929i)7-s + (0.634 + 1.09i)8-s + (−2.27 − 1.90i)9-s + (−1.39 − 1.16i)10-s + (1.65 + 2.86i)11-s + (−0.115 + 0.199i)12-s + (−2.49 + 0.908i)13-s + (0.338 + 1.92i)14-s + (−0.0307 + 0.174i)15-s + (4.61 + 1.67i)16-s + (−3.06 + 2.57i)17-s + ⋯ |
| L(s) = 1 | + (0.984 − 0.825i)2-s + (−0.0960 − 0.0349i)3-s + (0.113 − 0.640i)4-s + (−0.0776 − 0.440i)5-s + (−0.123 + 0.0449i)6-s + (−0.202 + 0.351i)7-s + (0.224 + 0.388i)8-s + (−0.758 − 0.636i)9-s + (−0.440 − 0.369i)10-s + (0.499 + 0.864i)11-s + (−0.0332 + 0.0576i)12-s + (−0.692 + 0.252i)13-s + (0.0905 + 0.513i)14-s + (−0.00794 + 0.0450i)15-s + (1.15 + 0.419i)16-s + (−0.743 + 0.623i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27811 - 0.659352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27811 - 0.659352i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.281 + 4.34i)T \) |
| good | 2 | \( 1 + (-1.39 + 1.16i)T + (0.347 - 1.96i)T^{2} \) |
| 3 | \( 1 + (0.166 + 0.0605i)T + (2.29 + 1.92i)T^{2} \) |
| 7 | \( 1 + (0.536 - 0.929i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.65 - 2.86i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.49 - 0.908i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (3.06 - 2.57i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.304 + 1.72i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.72 - 1.44i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.02 + 6.97i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 5.64T + 37T^{2} \) |
| 41 | \( 1 + (-0.842 - 0.306i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.47 + 8.38i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (4.82 + 4.04i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.590 - 3.34i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (1.13 - 0.955i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.38 + 13.5i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-9.85 - 8.26i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.91 - 10.8i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-2.45 - 0.892i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-6.17 - 2.24i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (1.34 - 2.33i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.742 - 0.270i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (14.3 - 12.0i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.62014733400590912242880258273, −12.54158244849082180986334504152, −12.02131400590241633957751766606, −11.07862741422864887148570776646, −9.598614430924895560314795763138, −8.475990007090862876754352694998, −6.69122218329571578511592018991, −5.19848701723404063539754378239, −4.05136389256782939654996941960, −2.41638702630867584058136809777,
3.28116182522103002132157802172, 4.83777180805322432156813881678, 6.00531651099704588600309328024, 7.03304647069584230940736907735, 8.268147007850430290504104339644, 9.945200310849460454713374339227, 11.08350778600180881529772781234, 12.25539285520457619945825186974, 13.64130776745634314115954452456, 14.04804349364596801880525668316
