| L(s) = 1 | + (−0.900 − 0.433i)2-s + (−0.122 + 1.64i)3-s + (0.623 + 0.781i)4-s + (0.955 + 0.294i)5-s + (0.822 − 1.42i)6-s + (1.51 + 2.62i)7-s + (−0.222 − 0.974i)8-s + (0.289 + 0.0436i)9-s + (−0.733 − 0.680i)10-s + (0.485 − 0.608i)11-s + (−1.35 + 0.926i)12-s + (0.728 − 0.676i)13-s + (−0.226 − 3.02i)14-s + (−0.601 + 1.53i)15-s + (−0.222 + 0.974i)16-s + (−4.00 + 1.23i)17-s + ⋯ |
| L(s) = 1 | + (−0.637 − 0.306i)2-s + (−0.0709 + 0.947i)3-s + (0.311 + 0.390i)4-s + (0.427 + 0.131i)5-s + (0.335 − 0.581i)6-s + (0.573 + 0.993i)7-s + (−0.0786 − 0.344i)8-s + (0.0965 + 0.0145i)9-s + (−0.231 − 0.215i)10-s + (0.146 − 0.183i)11-s + (−0.392 + 0.267i)12-s + (0.202 − 0.187i)13-s + (−0.0606 − 0.809i)14-s + (−0.155 + 0.395i)15-s + (−0.0556 + 0.243i)16-s + (−0.970 + 0.299i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.212 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.896117 + 0.722026i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.896117 + 0.722026i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (-0.955 - 0.294i)T \) |
| 43 | \( 1 + (-3.89 - 5.27i)T \) |
| good | 3 | \( 1 + (0.122 - 1.64i)T + (-2.96 - 0.447i)T^{2} \) |
| 7 | \( 1 + (-1.51 - 2.62i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.485 + 0.608i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-0.728 + 0.676i)T + (0.971 - 12.9i)T^{2} \) |
| 17 | \( 1 + (4.00 - 1.23i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-2.98 + 0.449i)T + (18.1 - 5.60i)T^{2} \) |
| 23 | \( 1 + (0.127 + 0.325i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (-0.0110 - 0.147i)T + (-28.6 + 4.32i)T^{2} \) |
| 31 | \( 1 + (0.253 - 0.173i)T + (11.3 - 28.8i)T^{2} \) |
| 37 | \( 1 + (3.02 - 5.23i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.75 + 3.25i)T + (25.5 + 32.0i)T^{2} \) |
| 47 | \( 1 + (-6.48 - 8.12i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (4.43 + 4.11i)T + (3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (-0.470 + 2.06i)T + (-53.1 - 25.5i)T^{2} \) |
| 61 | \( 1 + (0.481 + 0.328i)T + (22.2 + 56.7i)T^{2} \) |
| 67 | \( 1 + (-4.68 + 0.706i)T + (64.0 - 19.7i)T^{2} \) |
| 71 | \( 1 + (-4.05 + 10.3i)T + (-52.0 - 48.2i)T^{2} \) |
| 73 | \( 1 + (-3.98 + 3.69i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (1.85 + 3.20i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.05 + 14.1i)T + (-82.0 - 12.3i)T^{2} \) |
| 89 | \( 1 + (0.147 - 1.96i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 + (3.12 - 3.92i)T + (-21.5 - 94.5i)T^{2} \) |
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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
−11.09687681342231592307900445403, −10.45825105667422678572663238691, −9.491472723217562751621317184613, −8.938907810391907811811972880783, −8.003293014847646127119501101670, −6.69062666707133417412155691558, −5.53572898955135561549947003627, −4.53677209242064103707406057587, −3.17517338295525418430105152721, −1.81705657423975134245986578913,
0.997693518429072975152272543746, 2.10333199673303209124929458780, 4.12962466074231113404350906196, 5.42872678212065982098841532178, 6.70925977317955104545874174334, 7.17274813634208826743665808207, 8.057834233887819019902861913366, 9.067106460633374440849272534676, 10.03195837794554101253168165332, 10.88805355891521410786457673808
