| L(s) = 1 | + (1.29 + 2.54i)2-s + (−0.363 − 2.29i)3-s + (−2.43 + 3.35i)4-s + (−4.45 − 2.26i)5-s + (5.36 − 3.89i)6-s + (−3.40 + 3.40i)7-s + (−0.414 − 0.0656i)8-s + (3.43 − 1.11i)9-s + (−0.00760 − 14.2i)10-s + (2.91 − 8.98i)11-s + (8.58 + 4.37i)12-s + (−11.3 + 22.3i)13-s + (−13.0 − 4.24i)14-s + (−3.58 + 11.0i)15-s + (4.75 + 14.6i)16-s + (1.24 − 7.84i)17-s + ⋯ |
| L(s) = 1 | + (0.647 + 1.27i)2-s + (−0.121 − 0.764i)3-s + (−0.609 + 0.838i)4-s + (−0.891 − 0.453i)5-s + (0.893 − 0.649i)6-s + (−0.486 + 0.486i)7-s + (−0.0518 − 0.00820i)8-s + (0.381 − 0.123i)9-s + (−0.000760 − 1.42i)10-s + (0.265 − 0.816i)11-s + (0.715 + 0.364i)12-s + (−0.876 + 1.72i)13-s + (−0.933 − 0.303i)14-s + (−0.238 + 0.736i)15-s + (0.297 + 0.914i)16-s + (0.0730 − 0.461i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.986215 + 0.464719i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.986215 + 0.464719i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (4.45 + 2.26i)T \) |
| good | 2 | \( 1 + (-1.29 - 2.54i)T + (-2.35 + 3.23i)T^{2} \) |
| 3 | \( 1 + (0.363 + 2.29i)T + (-8.55 + 2.78i)T^{2} \) |
| 7 | \( 1 + (3.40 - 3.40i)T - 49iT^{2} \) |
| 11 | \( 1 + (-2.91 + 8.98i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (11.3 - 22.3i)T + (-99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (-1.24 + 7.84i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (8.55 + 11.7i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-13.3 + 6.82i)T + (310. - 427. i)T^{2} \) |
| 29 | \( 1 + (7.08 - 9.74i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-36.2 + 26.3i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (20.9 + 10.6i)T + (804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-1.06 - 3.28i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (-7.21 - 7.21i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-7.37 + 1.16i)T + (2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (4.72 + 29.8i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (62.8 - 20.4i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (13.4 - 41.4i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-9.10 + 57.4i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (38.6 + 28.1i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (69.7 - 35.5i)T + (3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (56.8 - 78.2i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-74.5 - 11.8i)T + (6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (-25.3 - 8.24i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (-111. + 17.7i)T + (8.94e3 - 2.90e3i)T^{2} \) |
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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
−17.04883372391059419884042850899, −16.21181012804859854910853128803, −15.22615939462663228228842845226, −13.91531891321560894789605241634, −12.74591576709196192202365508027, −11.64837465478610724095251060575, −8.967086342922724995639500963396, −7.40107688435535323743336073498, −6.43325653369015211920099409710, −4.51273037196684539952811904056,
3.36497639082128903830892457031, 4.65362743079693007054633244275, 7.48095794572994976164551870167, 10.11406665083115928183037187387, 10.54840289409507834975180636521, 12.10923608091207119796942902201, 12.99284224951576566904974249091, 14.74285460053723998107612859772, 15.68637261975994400321605926893, 17.18678732178470667827805944540
