L(s) = 1 | + (−0.309 − 0.951i)2-s + (1.88 − 1.37i)3-s + (−0.809 + 0.587i)4-s + (1.06 + 1.96i)5-s + (−1.88 − 1.37i)6-s + 7-s + (0.809 + 0.587i)8-s + (0.753 − 2.32i)9-s + (1.54 − 1.61i)10-s + (−0.266 − 0.819i)11-s + (−0.720 + 2.21i)12-s + (1.36 − 4.20i)13-s + (−0.309 − 0.951i)14-s + (4.70 + 2.25i)15-s + (0.309 − 0.951i)16-s + (3.22 + 2.34i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (1.08 − 0.791i)3-s + (−0.404 + 0.293i)4-s + (0.475 + 0.879i)5-s + (−0.770 − 0.559i)6-s + 0.377·7-s + (0.286 + 0.207i)8-s + (0.251 − 0.773i)9-s + (0.487 − 0.511i)10-s + (−0.0802 − 0.247i)11-s + (−0.208 + 0.640i)12-s + (0.379 − 1.16i)13-s + (−0.0825 − 0.254i)14-s + (1.21 + 0.582i)15-s + (0.0772 − 0.237i)16-s + (0.783 + 0.568i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50225 - 0.969558i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50225 - 0.969558i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (-1.06 - 1.96i)T \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + (-1.88 + 1.37i)T + (0.927 - 2.85i)T^{2} \) |
| 11 | \( 1 + (0.266 + 0.819i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.36 + 4.20i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-3.22 - 2.34i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.11 + 0.813i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.705 + 2.17i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.564 - 0.409i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (6.02 + 4.37i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.65 - 8.16i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.109 - 0.336i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 6.63T + 43T^{2} \) |
| 47 | \( 1 + (4.27 - 3.10i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.878 - 0.637i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.30 + 10.1i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.54 - 10.9i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (0.460 + 0.334i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-5.14 + 3.74i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.28 + 10.1i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (10.2 - 7.43i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-11.0 - 8.02i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.71 - 11.4i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (11.2 - 8.14i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.14763298940447592263218471389, −10.44545067465816468748821983986, −9.524666958780445328888125312601, −8.313968724133743335146634870991, −7.909415969456362381389915163360, −6.74424588234842130844048084768, −5.46922859888285608797913518278, −3.56659173516356870593427831488, −2.74874701712635588276216955353, −1.60473654120970188349747417721,
1.83721228158804358818286647237, 3.68716364437585326430239041814, 4.68518015270830064852444223838, 5.65251000044644006267709687323, 7.08019533980134991502800126339, 8.206446370207272275198805954413, 8.903391141004013719575217148272, 9.480677272644898780325064698145, 10.26144838775700345978780814881, 11.63157297862735395311967874640