L(s) = 1 | + (0.253 + 1.44i)2-s + (−1.15 + 0.970i)3-s + (−0.130 + 0.0473i)4-s + (−1.69 − 1.41i)6-s + (2.03 − 3.52i)7-s + (1.36 + 2.35i)8-s + (−0.124 + 0.707i)9-s + (0.310 + 0.537i)11-s + (0.104 − 0.180i)12-s + (3.90 + 3.27i)13-s + (5.59 + 2.03i)14-s + (−3.26 + 2.73i)16-s + (0.0462 + 0.262i)17-s − 1.05·18-s + (0.399 + 4.34i)19-s + ⋯ |
L(s) = 1 | + (0.179 + 1.01i)2-s + (−0.667 + 0.560i)3-s + (−0.0650 + 0.0236i)4-s + (−0.690 − 0.579i)6-s + (0.769 − 1.33i)7-s + (0.481 + 0.833i)8-s + (−0.0416 + 0.235i)9-s + (0.0936 + 0.162i)11-s + (0.0301 − 0.0522i)12-s + (1.08 + 0.908i)13-s + (1.49 + 0.544i)14-s + (−0.815 + 0.684i)16-s + (0.0112 + 0.0636i)17-s − 0.247·18-s + (0.0917 + 0.995i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.806882 + 1.29556i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.806882 + 1.29556i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-0.399 - 4.34i)T \) |
good | 2 | \( 1 + (-0.253 - 1.44i)T + (-1.87 + 0.684i)T^{2} \) |
| 3 | \( 1 + (1.15 - 0.970i)T + (0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-2.03 + 3.52i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.310 - 0.537i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.90 - 3.27i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.0462 - 0.262i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (5.48 - 1.99i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.708 + 4.01i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 8.83T + 37T^{2} \) |
| 41 | \( 1 + (3.43 - 2.88i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.69 + 0.615i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (2.00 - 11.3i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (2.37 - 0.862i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-0.154 - 0.876i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (2.03 - 0.742i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.44 + 13.8i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.54 - 0.563i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (1.37 - 1.15i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-3.94 + 3.30i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.89 + 11.9i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.000572 + 0.000480i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.80 - 10.2i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.21878009467011147864849437951, −10.55787409350004057009744431155, −9.658977124342144041606733549641, −7.972971704984196060688609453392, −7.83750625949449203820397495947, −6.44071214359331156877990616973, −5.87273674702709286340507311844, −4.59828631302197015383105260391, −4.10677387857238847106870005266, −1.69354631658320615048801074865,
1.09050703385577844778739219120, 2.37032538451741711355633480771, 3.51220296086303505427511386873, 5.01558772094672058835113507960, 5.97957523299555604749898213010, 6.83273263405043331131921440985, 8.157986899541695023883079513273, 8.943541120129054638400124798427, 10.17151962696093267307347579609, 11.13537996624250194585558962664