| L(s) = 1 | + (0.821 + 0.299i)2-s + (1.69 − 0.975i)3-s + (−0.946 − 0.793i)4-s + (−1.17 + 2.52i)5-s + (1.68 − 0.296i)6-s + (−2.75 − 0.738i)7-s + (−1.41 − 2.45i)8-s + (0.404 − 0.700i)9-s + (−1.71 + 1.71i)10-s + (5.50 + 2.56i)11-s + (−2.37 − 0.418i)12-s + (−1.65 + 2.36i)13-s + (−2.04 − 1.43i)14-s + (0.472 + 5.40i)15-s + (−0.000787 − 0.00446i)16-s + (−1.42 − 5.31i)17-s + ⋯ |
| L(s) = 1 | + (0.581 + 0.211i)2-s + (0.975 − 0.563i)3-s + (−0.473 − 0.396i)4-s + (−0.525 + 1.12i)5-s + (0.686 − 0.121i)6-s + (−1.04 − 0.279i)7-s + (−0.500 − 0.866i)8-s + (0.134 − 0.233i)9-s + (−0.543 + 0.543i)10-s + (1.65 + 0.773i)11-s + (−0.685 − 0.120i)12-s + (−0.459 + 0.656i)13-s + (−0.546 − 0.382i)14-s + (0.122 + 1.39i)15-s + (−0.000196 − 0.00111i)16-s + (−0.345 − 1.28i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0785i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23420 - 0.0485656i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23420 - 0.0485656i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 73 | \( 1 + (8.24 + 2.24i)T \) |
| good | 2 | \( 1 + (-0.821 - 0.299i)T + (1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (-1.69 + 0.975i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.17 - 2.52i)T + (-3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (2.75 + 0.738i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-5.50 - 2.56i)T + (7.07 + 8.42i)T^{2} \) |
| 13 | \( 1 + (1.65 - 2.36i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (1.42 + 5.31i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.54 + 4.24i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (1.79 + 4.93i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.620 - 1.33i)T + (-18.6 + 22.2i)T^{2} \) |
| 31 | \( 1 + (-0.306 - 0.0268i)T + (30.5 + 5.38i)T^{2} \) |
| 37 | \( 1 + (-2.31 + 0.841i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-0.233 + 1.32i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (2.74 - 10.2i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (-2.08 + 1.46i)T + (16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-2.55 + 1.19i)T + (34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (-4.49 - 3.15i)T + (20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (2.38 + 0.419i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.909 - 0.160i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-10.6 - 3.86i)T + (54.3 + 45.6i)T^{2} \) |
| 79 | \( 1 + (13.0 - 2.30i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (1.25 - 1.25i)T - 83iT^{2} \) |
| 89 | \( 1 + (0.322 + 1.82i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (12.6 + 7.30i)T + (48.5 + 84.0i)T^{2} \) |
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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
−14.39248529729406871919006896646, −13.87853091637758776912209693732, −12.77092349115815221241832873275, −11.52309742046328625846601689519, −9.810972341761721481377968920845, −9.046598131967133607136391020077, −7.03287014662220698218188832138, −6.73503441212337584509195091964, −4.31644369692980740638039233183, −2.95426398774292922070422870349,
3.44387447165399885036887464739, 4.06437597560727783442734851301, 5.86986740402851655076711870443, 8.206840352344779317767204276433, 8.871185052984721708840113841741, 9.721734877483588194024792342236, 11.84957955979696412081828849942, 12.51213107948690113517001779129, 13.53771361981985306997943422534, 14.51235526071755028286342998448
