| L(s) = 1 | + (−1.78 + 0.282i)2-s + (0.0717 − 1.73i)3-s + (1.19 − 0.387i)4-s + (−2.23 − 0.124i)5-s + (0.360 + 3.10i)6-s + (−2.65 − 2.65i)7-s + (1.19 − 0.611i)8-s + (−2.98 − 0.248i)9-s + (4.01 − 0.408i)10-s + (1.51 − 2.08i)11-s + (−0.584 − 2.09i)12-s + (0.0649 − 0.410i)13-s + (5.47 + 3.97i)14-s + (−0.375 + 3.85i)15-s + (−3.99 + 2.90i)16-s + (0.931 + 1.82i)17-s + ⋯ |
| L(s) = 1 | + (−1.25 + 0.199i)2-s + (0.0414 − 0.999i)3-s + (0.596 − 0.193i)4-s + (−0.998 − 0.0556i)5-s + (0.147 + 1.26i)6-s + (−1.00 − 1.00i)7-s + (0.424 − 0.216i)8-s + (−0.996 − 0.0827i)9-s + (1.26 − 0.129i)10-s + (0.456 − 0.627i)11-s + (−0.168 − 0.603i)12-s + (0.0180 − 0.113i)13-s + (1.46 + 1.06i)14-s + (−0.0969 + 0.995i)15-s + (−0.998 + 0.725i)16-s + (0.226 + 0.443i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.558 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.152728 - 0.286974i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.152728 - 0.286974i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0717 + 1.73i)T \) |
| 5 | \( 1 + (2.23 + 0.124i)T \) |
| good | 2 | \( 1 + (1.78 - 0.282i)T + (1.90 - 0.618i)T^{2} \) |
| 7 | \( 1 + (2.65 + 2.65i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.51 + 2.08i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.0649 + 0.410i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-0.931 - 1.82i)T + (-9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-4.65 - 1.51i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (0.345 + 2.18i)T + (-21.8 + 7.10i)T^{2} \) |
| 29 | \( 1 + (1.43 + 4.42i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.77 + 8.53i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (4.34 + 0.688i)T + (35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (2.81 + 3.87i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (2.89 - 2.89i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.60 - 2.34i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.216 + 0.424i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (4.95 - 3.59i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.67 + 3.39i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-9.21 + 4.69i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (-12.9 + 4.19i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.06 - 0.168i)T + (69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (4.39 - 1.42i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (1.01 - 0.515i)T + (48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-5.82 - 4.23i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.522 + 1.02i)T + (-57.0 - 78.4i)T^{2} \) |
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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
−13.99501808165566327261806919104, −13.09967154267066863523772457594, −11.88121966348999718882866202150, −10.72618109680542433744620225202, −9.449608648871681858761198425407, −8.181565213265337468349276709772, −7.46704248469194122044963230531, −6.42611389926385130238309668027, −3.68577392639030845902285213201, −0.65515493934572741816136538684,
3.21972881888597262583545391539, 5.02674645601132133031546621745, 7.07770364085668980166755945122, 8.564090366611687946004094679838, 9.302457762426584285820042906822, 10.14907254904772325053753234550, 11.41036414661679116214592673386, 12.23805609568333251162969225464, 14.11560999176198662261012356634, 15.46167960023205611149259199817
