| L(s) = 1 | + (1.59 + 1.16i)2-s + (0.217 + 2.07i)3-s + (0.589 + 1.81i)4-s + (−0.5 − 0.866i)5-s + (−2.06 + 3.56i)6-s + (−2.00 − 0.427i)7-s + (0.0555 − 0.170i)8-s + (−1.31 + 0.279i)9-s + (0.206 − 1.96i)10-s + (2.12 − 2.36i)11-s + (−3.63 + 1.61i)12-s + (0.147 + 0.0658i)13-s + (−2.71 − 3.01i)14-s + (1.68 − 1.22i)15-s + (3.37 − 2.45i)16-s + (1.58 + 1.76i)17-s + ⋯ |
| L(s) = 1 | + (1.13 + 0.821i)2-s + (0.125 + 1.19i)3-s + (0.294 + 0.907i)4-s + (−0.223 − 0.387i)5-s + (−0.841 + 1.45i)6-s + (−0.759 − 0.161i)7-s + (0.0196 − 0.0604i)8-s + (−0.437 + 0.0930i)9-s + (0.0653 − 0.621i)10-s + (0.642 − 0.713i)11-s + (−1.04 + 0.467i)12-s + (0.0410 + 0.0182i)13-s + (−0.726 − 0.806i)14-s + (0.435 − 0.316i)15-s + (0.844 − 0.613i)16-s + (0.385 + 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0764 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0764 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26173 + 1.36215i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26173 + 1.36215i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (0.429 + 5.55i)T \) |
| good | 2 | \( 1 + (-1.59 - 1.16i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.217 - 2.07i)T + (-2.93 + 0.623i)T^{2} \) |
| 7 | \( 1 + (2.00 + 0.427i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (-2.12 + 2.36i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.147 - 0.0658i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.58 - 1.76i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (5.60 - 2.49i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (1.32 - 4.09i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (5.97 + 4.33i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-3.31 + 5.74i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.327 + 3.11i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-1.07 + 0.480i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (8.10 - 5.88i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (1.92 - 0.409i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-0.452 - 4.30i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + (-0.556 - 0.963i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.96 - 1.69i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (-3.32 + 3.68i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-10.0 - 11.1i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (1.03 - 9.84i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (3.34 + 10.2i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-4.60 - 14.1i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−13.30803884011729184401662998739, −12.62638964033484532474161249436, −11.28840351019913890326985951038, −10.04117375162525891433386630471, −9.237003930551888578776981686968, −7.85138513746453738524807956384, −6.38871503770719345953960005813, −5.52745497626437831083129799742, −4.07923300821539790714713822538, −3.70755163630027633261332185257,
1.96404179790528888788191400652, 3.21928279868604847816437299930, 4.61654317911799229180447919788, 6.28680758812828708842246537631, 7.05363338095207310536465425855, 8.409500292306651971965670746452, 9.935903585496242579097040868815, 11.13470906825842735532787352038, 12.10341005507577746916228174556, 12.73156312861453295134159272914
