| L(s) = 1 | + (4.89 − 2.82i)2-s + (26.2 − 15.1i)3-s + (15.9 − 27.7i)4-s + (−88.5 − 153. i)5-s + (85.5 − 148. i)6-s − 258.·7-s − 181. i·8-s + (93.3 − 161. i)9-s + (−867. − 500. i)10-s + 794.·11-s − 968. i·12-s + (2.16e3 + 1.25e3i)13-s + (−1.26e3 + 729. i)14-s + (−4.64e3 − 2.67e3i)15-s + (−512. − 886. i)16-s + (−2.02e3 − 3.51e3i)17-s + ⋯ |
| L(s) = 1 | + (0.612 − 0.353i)2-s + (0.970 − 0.560i)3-s + (0.249 − 0.433i)4-s + (−0.708 − 1.22i)5-s + (0.396 − 0.686i)6-s − 0.752·7-s − 0.353i·8-s + (0.128 − 0.221i)9-s + (−0.867 − 0.500i)10-s + 0.597·11-s − 0.560i·12-s + (0.985 + 0.569i)13-s + (−0.460 + 0.265i)14-s + (−1.37 − 0.794i)15-s + (−0.125 − 0.216i)16-s + (−0.412 − 0.715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.48079 - 2.22607i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48079 - 2.22607i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4.89 + 2.82i)T \) |
| 19 | \( 1 + (-6.68e3 + 1.52e3i)T \) |
| good | 3 | \( 1 + (-26.2 + 15.1i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (88.5 + 153. i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + 258.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 794.T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-2.16e3 - 1.25e3i)T + (2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + (2.02e3 + 3.51e3i)T + (-1.20e7 + 2.09e7i)T^{2} \) |
| 23 | \( 1 + (-9.30e3 + 1.61e4i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.33e4 - 1.34e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + 3.04e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 2.81e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (5.70e4 - 3.29e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-6.95e4 - 1.20e5i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-2.28e4 + 3.95e4i)T + (-5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (1.14e5 + 6.60e4i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (6.98e4 - 4.03e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-9.28e4 + 1.60e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (3.08e5 + 1.78e5i)T + (4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (1.95e5 - 1.12e5i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-3.62e5 - 6.28e5i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (4.81e5 - 2.77e5i)T + (1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 7.58e4T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.00e6 - 5.77e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (1.50e5 - 8.69e4i)T + (4.16e11 - 7.21e11i)T^{2} \) |
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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
−14.30620573647449519373850058614, −13.35806406475588038337206977931, −12.55080405673422421997813564962, −11.39044334035896065429635933796, −9.308637918185952067790397719107, −8.393021523341110395213332269667, −6.74005412718578277199792596966, −4.64676337116917892737916390871, −3.11143511058334911674041873629, −1.10286054525131288642926118236,
3.15008634922153922895307914002, 3.74396135712311811026062760619, 6.21932357759273904576122890898, 7.54791636078728686780249727182, 8.990342721211148472636301272060, 10.50066826601909573427773166881, 11.80963106555191701725047289716, 13.44404044789303773423549352264, 14.38572330530493223575599497407, 15.40985551731171376742346442304
