| L(s) = 1 | + (−0.281 + 2.67i)2-s + (1.33 + 1.10i)3-s + (−5.14 − 1.09i)4-s + (1.43 − 0.151i)5-s + (−3.33 + 3.26i)6-s + (−1.07 − 0.965i)7-s + (2.71 − 8.34i)8-s + (0.560 + 2.94i)9-s + 3.89i·10-s + (2.66 − 1.96i)11-s + (−5.65 − 7.13i)12-s + (−1.21 − 2.71i)13-s + (2.89 − 2.60i)14-s + (2.08 + 1.38i)15-s + (11.9 + 5.33i)16-s + (−3.35 + 2.44i)17-s + ⋯ |
| L(s) = 1 | + (−0.199 + 1.89i)2-s + (0.770 + 0.637i)3-s + (−2.57 − 0.546i)4-s + (0.643 − 0.0676i)5-s + (−1.36 + 1.33i)6-s + (−0.405 − 0.365i)7-s + (0.958 − 2.95i)8-s + (0.186 + 0.982i)9-s + 1.23i·10-s + (0.804 − 0.593i)11-s + (−1.63 − 2.06i)12-s + (−0.335 − 0.754i)13-s + (0.772 − 0.695i)14-s + (0.538 + 0.358i)15-s + (2.99 + 1.33i)16-s + (−0.814 + 0.591i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.277404 + 1.00753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.277404 + 1.00753i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.33 - 1.10i)T \) |
| 11 | \( 1 + (-2.66 + 1.96i)T \) |
| good | 2 | \( 1 + (0.281 - 2.67i)T + (-1.95 - 0.415i)T^{2} \) |
| 5 | \( 1 + (-1.43 + 0.151i)T + (4.89 - 1.03i)T^{2} \) |
| 7 | \( 1 + (1.07 + 0.965i)T + (0.731 + 6.96i)T^{2} \) |
| 13 | \( 1 + (1.21 + 2.71i)T + (-8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (3.35 - 2.44i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (1.19 + 0.387i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-5.02 - 2.90i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0651 - 0.0723i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (-3.73 + 1.66i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (3.08 + 9.49i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (3.05 + 3.39i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (1.84 - 1.06i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.50 - 7.10i)T + (-42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (0.197 - 0.271i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.629 + 2.96i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-0.129 + 0.290i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (1.71 - 2.97i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.38 + 7.41i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (6.75 - 2.19i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-9.96 - 1.04i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-5.44 - 2.42i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 - 5.11iT - 89T^{2} \) |
| 97 | \( 1 + (0.631 - 6.00i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−14.59399629236614239459089042614, −13.66326898684273096461312948678, −13.10279516049642928976922757687, −10.54124613022602393682704519081, −9.454281637617961927763413057525, −8.815884749923320837342808635121, −7.67411580528085226967702916252, −6.45697645839098973403279851108, −5.27599136831934172349154275498, −3.86632840649742537987645302686,
1.77832704596889595801477344588, 2.90652296113929338944558796235, 4.47770103746364499974390816826, 6.73364644246171017932868710913, 8.618300926594450074898276457376, 9.327802265049584871430318954232, 10.08620536324166301270698851738, 11.62062150417096126986067464261, 12.28831808327972017573386686091, 13.29913784900603257675174829769
