| L(s) = 1 | + (0.222 + 0.974i)2-s + (1.79 + 1.66i)3-s + (−0.900 + 0.433i)4-s + (−0.988 + 0.149i)5-s + (−1.22 + 2.11i)6-s + (0.576 + 0.999i)7-s + (−0.623 − 0.781i)8-s + (0.222 + 2.97i)9-s + (−0.365 − 0.930i)10-s + (0.488 + 0.235i)11-s + (−2.33 − 0.720i)12-s + (−1.72 + 4.39i)13-s + (−0.845 + 0.784i)14-s + (−2.02 − 1.37i)15-s + (0.623 − 0.781i)16-s + (2.55 + 0.385i)17-s + ⋯ |
| L(s) = 1 | + (0.157 + 0.689i)2-s + (1.03 + 0.960i)3-s + (−0.450 + 0.216i)4-s + (−0.442 + 0.0666i)5-s + (−0.499 + 0.864i)6-s + (0.218 + 0.377i)7-s + (−0.220 − 0.276i)8-s + (0.0742 + 0.990i)9-s + (−0.115 − 0.294i)10-s + (0.147 + 0.0709i)11-s + (−0.674 − 0.208i)12-s + (−0.477 + 1.21i)13-s + (−0.226 + 0.209i)14-s + (−0.521 − 0.355i)15-s + (0.155 − 0.195i)16-s + (0.620 + 0.0935i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.679673 + 1.67907i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.679673 + 1.67907i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.222 - 0.974i)T \) |
| 5 | \( 1 + (0.988 - 0.149i)T \) |
| 43 | \( 1 + (-6.30 - 1.78i)T \) |
| good | 3 | \( 1 + (-1.79 - 1.66i)T + (0.224 + 2.99i)T^{2} \) |
| 7 | \( 1 + (-0.576 - 0.999i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.488 - 0.235i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (1.72 - 4.39i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (-2.55 - 0.385i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.184 + 2.45i)T + (-18.7 - 2.83i)T^{2} \) |
| 23 | \( 1 + (6.89 - 4.69i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (-6.68 + 6.19i)T + (2.16 - 28.9i)T^{2} \) |
| 31 | \( 1 + (-1.17 - 0.362i)T + (25.6 + 17.4i)T^{2} \) |
| 37 | \( 1 + (-0.736 + 1.27i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.46 + 6.41i)T + (-36.9 + 17.7i)T^{2} \) |
| 47 | \( 1 + (-7.76 + 3.73i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (-0.276 - 0.704i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (2.63 - 3.30i)T + (-13.1 - 57.5i)T^{2} \) |
| 61 | \( 1 + (-0.186 + 0.0576i)T + (50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (0.268 - 3.58i)T + (-66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (2.67 + 1.82i)T + (25.9 + 66.0i)T^{2} \) |
| 73 | \( 1 + (4.56 - 11.6i)T + (-53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-6.22 - 10.7i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.30 + 3.06i)T + (6.20 + 82.7i)T^{2} \) |
| 89 | \( 1 + (3.79 + 3.51i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + (-5.14 - 2.47i)T + (60.4 + 75.8i)T^{2} \) |
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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
−11.65384751929573746655956243154, −10.22835194679295716253039508347, −9.503762204295336866460610600164, −8.751365127552765248066910921776, −7.979477883445414923000317007130, −7.02580103905253039934842995217, −5.72002893876101258554332877646, −4.44796808193666347885074837260, −3.86750778037886765842959573427, −2.48728820250815179091653028859,
1.07696119853305061841451855537, 2.52985696290006506398361656717, 3.44803909599774211936343846668, 4.71311306488291956011398950221, 6.14938139496713875657416046451, 7.52290021817751775378537906590, 8.009163136667260395190364377966, 8.831294212148694661059603492436, 10.07171543375316269844425960368, 10.73341665077892234630488580925
