| L(s) = 1 | + (−0.156 − 0.987i)2-s + (−2.69 − 1.37i)3-s + (−0.951 + 0.309i)4-s + (2.18 − 0.485i)5-s + (−0.934 + 2.87i)6-s + (−0.391 + 0.391i)7-s + (0.453 + 0.891i)8-s + (3.60 + 4.96i)9-s + (−0.820 − 2.07i)10-s + (1.05 + 0.769i)11-s + (2.98 + 0.472i)12-s + (0.269 − 1.69i)13-s + (0.447 + 0.325i)14-s + (−6.54 − 1.68i)15-s + (0.809 − 0.587i)16-s + (−1.36 − 2.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 − 0.698i)2-s + (−1.55 − 0.792i)3-s + (−0.475 + 0.154i)4-s + (0.976 − 0.217i)5-s + (−0.381 + 1.17i)6-s + (−0.147 + 0.147i)7-s + (0.160 + 0.315i)8-s + (1.20 + 1.65i)9-s + (−0.259 − 0.657i)10-s + (0.319 + 0.232i)11-s + (0.861 + 0.136i)12-s + (0.0746 − 0.471i)13-s + (0.119 + 0.0869i)14-s + (−1.68 − 0.435i)15-s + (0.202 − 0.146i)16-s + (−0.330 − 0.648i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0644277 + 0.653320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0644277 + 0.653320i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.156 + 0.987i)T \) |
| 5 | \( 1 + (-2.18 + 0.485i)T \) |
| 19 | \( 1 + (1.19 + 4.19i)T \) |
| good | 3 | \( 1 + (2.69 + 1.37i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (0.391 - 0.391i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.05 - 0.769i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.269 + 1.69i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (1.36 + 2.67i)T + (-9.99 + 13.7i)T^{2} \) |
| 23 | \( 1 + (1.08 + 6.87i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (-1.40 - 4.33i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (6.23 + 2.02i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-8.53 - 1.35i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (1.54 + 2.13i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-2.90 - 2.90i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.54 + 3.33i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (12.5 + 6.38i)T + (31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (1.79 - 1.30i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (9.64 + 7.00i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (6.92 - 3.52i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-10.1 + 3.28i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.25 + 7.89i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (-2.36 - 7.28i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.51 + 1.28i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-12.1 - 8.83i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.07 + 8.00i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−9.742867231399077251191787843580, −9.052148087922867685992004763204, −7.84286518527751599394463313647, −6.69101363339602804561094844267, −6.23154031125577482300603433081, −5.17308664799267537514851871152, −4.60210296016365018850713089799, −2.69898557322420006423343418221, −1.61174737604596102326689458320, −0.41269393667064748274386434403,
1.49616171652096118615575639427, 3.67226974638432286751283007786, 4.57792810558589430134941616512, 5.58478365653399858177060871209, 6.09294890694261915146428456155, 6.63583459435370576539154757047, 7.80856206181972970697330247426, 9.209645750855462972252722047265, 9.631776810801720650827378468678, 10.45707218632086348399190712707
