| L(s) = 1 | + (0.987 + 0.156i)2-s + (1.66 − 0.846i)3-s + (0.951 + 0.309i)4-s + (0.499 + 2.17i)5-s + (1.77 − 0.576i)6-s + (−2.54 − 0.737i)7-s + (0.891 + 0.453i)8-s + (0.281 − 0.387i)9-s + (0.152 + 2.23i)10-s + (4.96 − 3.60i)11-s + (1.84 − 0.291i)12-s + (0.692 + 4.37i)13-s + (−2.39 − 1.12i)14-s + (2.67 + 3.19i)15-s + (0.809 + 0.587i)16-s + (−1.52 − 0.774i)17-s + ⋯ |
| L(s) = 1 | + (0.698 + 0.110i)2-s + (0.959 − 0.488i)3-s + (0.475 + 0.154i)4-s + (0.223 + 0.974i)5-s + (0.724 − 0.235i)6-s + (−0.960 − 0.278i)7-s + (0.315 + 0.160i)8-s + (0.0937 − 0.129i)9-s + (0.0480 + 0.705i)10-s + (1.49 − 1.08i)11-s + (0.531 − 0.0842i)12-s + (0.192 + 1.21i)13-s + (−0.639 − 0.301i)14-s + (0.690 + 0.826i)15-s + (0.202 + 0.146i)16-s + (−0.368 − 0.187i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.54764 + 0.173715i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.54764 + 0.173715i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.987 - 0.156i)T \) |
| 5 | \( 1 + (-0.499 - 2.17i)T \) |
| 7 | \( 1 + (2.54 + 0.737i)T \) |
| good | 3 | \( 1 + (-1.66 + 0.846i)T + (1.76 - 2.42i)T^{2} \) |
| 11 | \( 1 + (-4.96 + 3.60i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.692 - 4.37i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (1.52 + 0.774i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (1.94 + 5.99i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.284 + 1.79i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (8.17 + 2.65i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (4.43 - 1.43i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.563 + 3.55i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.248 + 0.341i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-0.714 + 0.714i)T - 43iT^{2} \) |
| 47 | \( 1 + (-0.635 - 1.24i)T + (-27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-4.87 - 9.57i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (10.6 + 7.70i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.203 - 0.280i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.457 + 0.898i)T + (-39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (2.47 - 7.60i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-11.7 - 1.85i)T + (69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-2.27 - 0.739i)T + (63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.48 + 6.84i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (2.13 - 1.55i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (2.22 + 4.37i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−11.37708460884883478299008401565, −10.96235319553307607464689057942, −9.331197695158632660513871577013, −8.899745361914404388926818357834, −7.33108393484241722271761027889, −6.74487522107121318848114776921, −6.00409744679432349242650013464, −4.04141577045533733876219734743, −3.24100625574946252283106412815, −2.12917047577723045033550285459,
1.88006795313299471145873247767, 3.48659202607677007244282983380, 4.07520610031249672490983730518, 5.49620163711474126558254945470, 6.42174343131608671526132284629, 7.80777187573156414292901983281, 8.957497411919570817984211360088, 9.492251437572597683078916075575, 10.29621693194480826164589519430, 11.82339586225989163600764363734
