L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (3.19 − 1.84i)5-s + (−1.52 + 2.15i)7-s − 0.999·8-s + (3.19 + 1.84i)10-s + (−0.412 + 3.29i)11-s + 5.60i·13-s + (−2.63 − 0.245i)14-s + (−0.5 − 0.866i)16-s + (1.39 − 2.41i)17-s + (−2.14 + 1.23i)19-s + 3.69i·20-s + (−3.05 + 1.28i)22-s + (−3.26 + 1.88i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (1.42 − 0.825i)5-s + (−0.578 + 0.815i)7-s − 0.353·8-s + (1.01 + 0.583i)10-s + (−0.124 + 0.992i)11-s + 1.55i·13-s + (−0.704 − 0.0656i)14-s + (−0.125 − 0.216i)16-s + (0.338 − 0.585i)17-s + (−0.492 + 0.284i)19-s + 0.825i·20-s + (−0.651 + 0.274i)22-s + (−0.680 + 0.392i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.129965344\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.129965344\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1.52 - 2.15i)T \) |
| 11 | \( 1 + (0.412 - 3.29i)T \) |
good | 5 | \( 1 + (-3.19 + 1.84i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 5.60iT - 13T^{2} \) |
| 17 | \( 1 + (-1.39 + 2.41i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.14 - 1.23i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.26 - 1.88i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 1.51T + 29T^{2} \) |
| 31 | \( 1 + (1.70 - 2.94i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.85 - 4.95i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 + 7.72iT - 43T^{2} \) |
| 47 | \( 1 + (8.98 - 5.18i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.27 - 5.35i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-11.6 - 6.71i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.13 - 1.23i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.66 + 9.81i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.15iT - 71T^{2} \) |
| 73 | \( 1 + (4.53 + 2.62i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.250 - 0.144i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + (13.7 - 7.93i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.58T + 97T^{2} \) |
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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
−9.546777880162924217311538078355, −9.148274680537291038055232857425, −8.359830698883454241267608575561, −7.12048841507972959879280793414, −6.41185778418652544844412387633, −5.69811032772793814799504634233, −4.97686660756366397062592887604, −4.12532484664089111994493484431, −2.55536322459167095795191192746, −1.71489734005105618650324858768,
0.75697963649259923040995424106, 2.26108476509300524945777697954, 3.07763839824477409930026758987, 3.87746309700328794527164672908, 5.32081810450786199295629171272, 6.00363679125232139771942856887, 6.52507009172451503785161283783, 7.70751099601287887017671082837, 8.659275163372969532696101655963, 9.868084492756845344641007924200