| L(s) = 1 | + (0.984 + 0.173i)2-s + (0.307 − 1.70i)3-s + (0.939 + 0.342i)4-s + (2.17 + 0.533i)5-s + (0.598 − 1.62i)6-s + (0.260 − 0.150i)7-s + (0.866 + 0.5i)8-s + (−2.81 − 1.04i)9-s + (2.04 + 0.902i)10-s + (1.17 + 0.680i)11-s + (0.871 − 1.49i)12-s + (2.32 − 1.94i)13-s + (0.282 − 0.102i)14-s + (1.57 − 3.53i)15-s + (0.766 + 0.642i)16-s + (−0.0110 + 0.0629i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (0.177 − 0.984i)3-s + (0.469 + 0.171i)4-s + (0.971 + 0.238i)5-s + (0.244 − 0.663i)6-s + (0.0985 − 0.0569i)7-s + (0.306 + 0.176i)8-s + (−0.936 − 0.349i)9-s + (0.646 + 0.285i)10-s + (0.355 + 0.205i)11-s + (0.251 − 0.432i)12-s + (0.644 − 0.540i)13-s + (0.0756 − 0.0275i)14-s + (0.407 − 0.913i)15-s + (0.191 + 0.160i)16-s + (−0.00269 + 0.0152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.56777 - 0.819786i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.56777 - 0.819786i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 - 0.173i)T \) |
| 3 | \( 1 + (-0.307 + 1.70i)T \) |
| 5 | \( 1 + (-2.17 - 0.533i)T \) |
| 19 | \( 1 + (3.34 - 2.79i)T \) |
| good | 7 | \( 1 + (-0.260 + 0.150i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.17 - 0.680i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.32 + 1.94i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.0110 - 0.0629i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (0.311 + 0.113i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (1.30 + 7.39i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.02 - 1.16i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 + (4.00 + 3.35i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.86 - 5.13i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.167 - 0.949i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.00 + 5.49i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.12 + 6.40i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.91 - 1.06i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.99 - 11.3i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (7.92 - 2.88i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-6.48 + 7.72i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (9.29 - 11.0i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.598 + 1.03i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (10.2 - 8.62i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.21 + 6.90i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.82286717549100399738634032876, −9.838266904178724749658014431808, −8.700590372863239066985268255774, −7.85357674395622638083644349691, −6.78280775557002741645560281859, −6.14217902437143329389309014177, −5.38039273815845484688980584198, −3.84383000551365584698727421329, −2.60108158905042266404804691538, −1.54880202460225211151753434774,
1.86755706477461203508375016935, 3.16776299758057141854424775793, 4.25384752342657838363514288936, 5.15814678983584136644609997765, 5.98053715835330782631901965973, 6.92429596993858644842864511731, 8.589274495148315936059512927756, 9.062430182900208720227851419048, 10.07875416191113820572434612153, 10.81282771618496082567174350156
