| L(s) = 1 | + (−1.31 − 0.269i)2-s + (−0.987 + 0.160i)3-s + (−0.174 − 0.0741i)4-s + (0.381 − 1.54i)5-s + (1.34 + 0.0541i)6-s + (1.87 − 0.888i)7-s + (2.42 + 1.67i)8-s + (0.948 − 0.316i)9-s + (−0.919 + 1.93i)10-s + (0.329 − 0.986i)11-s + (0.183 + 0.0452i)12-s + (−2.35 + 2.72i)13-s + (−2.70 + 0.667i)14-s + (−0.128 + 1.58i)15-s + (−2.48 − 2.58i)16-s + (0.00216 + 0.00455i)17-s + ⋯ |
| L(s) = 1 | + (−0.932 − 0.190i)2-s + (−0.569 + 0.0926i)3-s + (−0.0870 − 0.0370i)4-s + (0.170 − 0.692i)5-s + (0.548 + 0.0221i)6-s + (0.707 − 0.335i)7-s + (0.857 + 0.591i)8-s + (0.316 − 0.105i)9-s + (−0.290 + 0.612i)10-s + (0.0992 − 0.297i)11-s + (0.0530 + 0.0130i)12-s + (−0.654 + 0.756i)13-s + (−0.723 + 0.178i)14-s + (−0.0331 + 0.410i)15-s + (−0.620 − 0.646i)16-s + (0.000524 + 0.00110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0202 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0202 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.474289 - 0.464784i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.474289 - 0.464784i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.987 - 0.160i)T \) |
| 13 | \( 1 + (2.35 - 2.72i)T \) |
| good | 2 | \( 1 + (1.31 + 0.269i)T + (1.83 + 0.783i)T^{2} \) |
| 5 | \( 1 + (-0.381 + 1.54i)T + (-4.42 - 2.32i)T^{2} \) |
| 7 | \( 1 + (-1.87 + 0.888i)T + (4.42 - 5.42i)T^{2} \) |
| 11 | \( 1 + (-0.329 + 0.986i)T + (-8.79 - 6.60i)T^{2} \) |
| 17 | \( 1 + (-0.00216 - 0.00455i)T + (-10.7 + 13.1i)T^{2} \) |
| 19 | \( 1 + (-6.24 - 3.60i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.22 + 5.58i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.80 + 8.84i)T + (-26.6 - 11.3i)T^{2} \) |
| 31 | \( 1 + (2.25 + 4.30i)T + (-17.6 + 25.5i)T^{2} \) |
| 37 | \( 1 + (-2.59 + 4.10i)T + (-15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (0.616 + 3.79i)T + (-38.8 + 12.9i)T^{2} \) |
| 43 | \( 1 + (-0.178 + 0.112i)T + (18.4 - 38.8i)T^{2} \) |
| 47 | \( 1 + (-1.11 - 0.135i)T + (45.6 + 11.2i)T^{2} \) |
| 53 | \( 1 + (-5.40 + 7.83i)T + (-18.7 - 49.5i)T^{2} \) |
| 59 | \( 1 + (3.57 + 3.43i)T + (2.37 + 58.9i)T^{2} \) |
| 61 | \( 1 + (1.23 - 0.0995i)T + (60.2 - 9.78i)T^{2} \) |
| 67 | \( 1 + (-0.760 - 1.78i)T + (-46.4 + 48.3i)T^{2} \) |
| 71 | \( 1 + (2.24 - 1.83i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (6.75 + 7.62i)T + (-8.79 + 72.4i)T^{2} \) |
| 79 | \( 1 + (-1.93 + 15.9i)T + (-76.7 - 18.9i)T^{2} \) |
| 83 | \( 1 + (12.4 - 4.72i)T + (62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (-6.98 + 4.03i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.6 - 3.08i)T + (81.9 + 51.8i)T^{2} \) |
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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
−10.50955611676443975681919876229, −9.776247428918449117355630409086, −9.077796166757081107213870130111, −8.096469900155528242178304466266, −7.39367191313430892513538026782, −5.94032518951501483815639714381, −4.94468399936268157578001639859, −4.19819077686749737643015948976, −1.94179477676489880628526522868, −0.68165635895347465970235966693,
1.33044820640277304932213318275, 3.06791154675421520746499517100, 4.70768348063282255037546172587, 5.51263606052788601954959727249, 6.97462680653071131700512423903, 7.43904319159749588958585647051, 8.435354404496571754311358472684, 9.444987865416333271937041953896, 10.16292891604879576600583732124, 10.92231521318249264505552508046
