| L(s) = 1 | + (0.415 − 0.909i)2-s + (−1.71 + 1.10i)3-s + (−0.654 − 0.755i)4-s + (0.142 − 0.989i)5-s + (0.290 + 2.01i)6-s + (−0.0216 + 0.0474i)7-s + (−0.959 + 0.281i)8-s + (0.483 − 1.05i)9-s + (−0.841 − 0.540i)10-s + (−0.279 + 1.94i)11-s + (1.95 + 0.574i)12-s + (3.68 + 1.08i)13-s + (0.0341 + 0.0393i)14-s + (0.847 + 1.85i)15-s + (−0.142 + 0.989i)16-s + (4.53 − 5.23i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (−0.991 + 0.636i)3-s + (−0.327 − 0.377i)4-s + (0.0636 − 0.442i)5-s + (0.118 + 0.824i)6-s + (−0.00818 + 0.0179i)7-s + (−0.339 + 0.0996i)8-s + (0.161 − 0.353i)9-s + (−0.266 − 0.170i)10-s + (−0.0841 + 0.585i)11-s + (0.565 + 0.165i)12-s + (1.02 + 0.300i)13-s + (0.00912 + 0.0105i)14-s + (0.218 + 0.479i)15-s + (−0.0355 + 0.247i)16-s + (1.10 − 1.27i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.11141 - 0.492042i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11141 - 0.492042i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.142 + 0.989i)T \) |
| 67 | \( 1 + (-0.539 + 8.16i)T \) |
| good | 3 | \( 1 + (1.71 - 1.10i)T + (1.24 - 2.72i)T^{2} \) |
| 7 | \( 1 + (0.0216 - 0.0474i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.279 - 1.94i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-3.68 - 1.08i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-4.53 + 5.23i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (1.51 + 3.32i)T + (-12.4 + 14.3i)T^{2} \) |
| 23 | \( 1 + (4.63 - 2.97i)T + (9.55 - 20.9i)T^{2} \) |
| 29 | \( 1 - 10.2T + 29T^{2} \) |
| 31 | \( 1 + (-9.45 + 2.77i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 - 0.657T + 37T^{2} \) |
| 41 | \( 1 + (-3.10 + 3.58i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-5.46 + 6.31i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (3.25 - 2.08i)T + (19.5 - 42.7i)T^{2} \) |
| 53 | \( 1 + (5.64 + 6.51i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-5.14 + 1.51i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-2.13 - 14.8i)T + (-58.5 + 17.1i)T^{2} \) |
| 71 | \( 1 + (-0.0674 - 0.0778i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.829 + 5.77i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (2.53 + 0.745i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (1.21 - 8.45i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (10.9 + 7.01i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + 10.5T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.36164424550476314736390092831, −9.902165070330808542960558893193, −8.955802484602077189966258238381, −7.87307500717522611350640828550, −6.49401295978888384186427095074, −5.61594813224811329908255303184, −4.79269628584760915842852555568, −4.09830820490351722222475127206, −2.62027854714107864443248106637, −0.885461740962675298277087681022,
1.11610640543360380690632820546, 3.10858711283808558455410936008, 4.25445754934335254162491357544, 5.67832463655976749734105832897, 6.16142900438752307338252478544, 6.67800595283309069868470173422, 8.106012899808469421583173332760, 8.339538377141371510943240159791, 10.01618289032127793934853584550, 10.64980111647187782555614636084
