| L(s) = 1 | + (0.415 − 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−2.63 + 1.69i)5-s + (−0.654 + 0.755i)6-s + (−0.142 + 0.989i)7-s + (−0.959 + 0.281i)8-s + (0.841 + 0.540i)9-s + (0.445 + 3.09i)10-s + (−0.711 − 1.55i)11-s + (0.415 + 0.909i)12-s + (0.176 + 1.22i)13-s + (0.841 + 0.540i)14-s + (3.00 − 0.881i)15-s + (−0.142 + 0.989i)16-s + (3.07 − 3.54i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (−1.17 + 0.756i)5-s + (−0.267 + 0.308i)6-s + (−0.0537 + 0.374i)7-s + (−0.339 + 0.0996i)8-s + (0.280 + 0.180i)9-s + (0.140 + 0.978i)10-s + (−0.214 − 0.469i)11-s + (0.119 + 0.262i)12-s + (0.0488 + 0.339i)13-s + (0.224 + 0.144i)14-s + (0.774 − 0.227i)15-s + (−0.0355 + 0.247i)16-s + (0.745 − 0.859i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.798796 - 0.640438i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.798796 - 0.640438i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 + 0.909i)T \) |
| 3 | \( 1 + (0.959 + 0.281i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (-0.438 + 4.77i)T \) |
| good | 5 | \( 1 + (2.63 - 1.69i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (0.711 + 1.55i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.176 - 1.22i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-3.07 + 3.54i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.37 - 2.74i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-4.79 + 5.53i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.842 + 0.247i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (7.08 + 4.55i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-2.26 + 1.45i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-7.88 - 2.31i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 9.80T + 47T^{2} \) |
| 53 | \( 1 + (-0.125 + 0.871i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.492 + 3.42i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (5.04 - 1.48i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-2.75 + 6.03i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (-2.50 + 5.47i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-6.11 - 7.05i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.350 + 2.43i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-4.53 - 2.91i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (6.92 + 2.03i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (-8.37 + 5.38i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.15865501632440096566051990487, −9.124304612914170860375004578111, −8.050401140044206935181092473699, −7.33354446753187461757187149866, −6.33574306396330939884022275399, −5.43346508284040555088082018537, −4.36373332285841581913256058455, −3.44024855602491112529802481514, −2.48832868006374530921700586800, −0.63252146530985972777153203417,
0.988388910735291261821757548972, 3.30556825005214853155376705325, 4.17925053272835751005716673480, 4.96747026465580945840836429891, 5.72020187150615309112533831121, 6.96149741412449985419500309057, 7.57875183178915630003293509366, 8.332421398379111956724402883596, 9.207464945562943198348993847348, 10.25319305973866994195550198592
