| L(s) = 1 | + (1.39 − 0.222i)2-s + (−1.60 − 0.559i)3-s + (1.90 − 0.621i)4-s + (1.69 + 0.386i)5-s + (−2.35 − 0.426i)6-s + (−0.260 − 0.541i)7-s + (2.51 − 1.29i)8-s + (−0.0983 − 0.0784i)9-s + (2.44 + 0.163i)10-s + (−0.116 + 1.03i)11-s + (−3.38 − 0.0706i)12-s + (−3.87 + 3.08i)13-s + (−0.484 − 0.697i)14-s + (−2.49 − 1.56i)15-s + (3.22 − 2.36i)16-s + (−1.06 + 1.06i)17-s + ⋯ |
| L(s) = 1 | + (0.987 − 0.157i)2-s + (−0.923 − 0.323i)3-s + (0.950 − 0.310i)4-s + (0.756 + 0.172i)5-s + (−0.963 − 0.174i)6-s + (−0.0984 − 0.204i)7-s + (0.889 − 0.456i)8-s + (−0.0327 − 0.0261i)9-s + (0.774 + 0.0516i)10-s + (−0.0350 + 0.310i)11-s + (−0.978 − 0.0204i)12-s + (−1.07 + 0.856i)13-s + (−0.129 − 0.186i)14-s + (−0.643 − 0.404i)15-s + (0.807 − 0.590i)16-s + (−0.257 + 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.42652 - 0.360032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.42652 - 0.360032i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.39 + 0.222i)T \) |
| 29 | \( 1 + (-1.22 + 5.24i)T \) |
| good | 3 | \( 1 + (1.60 + 0.559i)T + (2.34 + 1.87i)T^{2} \) |
| 5 | \( 1 + (-1.69 - 0.386i)T + (4.50 + 2.16i)T^{2} \) |
| 7 | \( 1 + (0.260 + 0.541i)T + (-4.36 + 5.47i)T^{2} \) |
| 11 | \( 1 + (0.116 - 1.03i)T + (-10.7 - 2.44i)T^{2} \) |
| 13 | \( 1 + (3.87 - 3.08i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (1.06 - 1.06i)T - 17iT^{2} \) |
| 19 | \( 1 + (-1.36 - 3.91i)T + (-14.8 + 11.8i)T^{2} \) |
| 23 | \( 1 + (7.10 - 1.62i)T + (20.7 - 9.97i)T^{2} \) |
| 31 | \( 1 + (-3.05 + 1.91i)T + (13.4 - 27.9i)T^{2} \) |
| 37 | \( 1 + (-5.84 + 0.658i)T + (36.0 - 8.23i)T^{2} \) |
| 41 | \( 1 + (3.60 + 3.60i)T + 41iT^{2} \) |
| 43 | \( 1 + (-4.74 + 7.54i)T + (-18.6 - 38.7i)T^{2} \) |
| 47 | \( 1 + (-0.738 - 0.0832i)T + (45.8 + 10.4i)T^{2} \) |
| 53 | \( 1 + (2.05 - 9.00i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + 1.92iT - 59T^{2} \) |
| 61 | \( 1 + (4.10 - 11.7i)T + (-47.6 - 38.0i)T^{2} \) |
| 67 | \( 1 + (0.940 - 1.17i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (7.06 + 8.85i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-2.10 + 3.34i)T + (-31.6 - 65.7i)T^{2} \) |
| 79 | \( 1 + (-16.2 + 1.83i)T + (77.0 - 17.5i)T^{2} \) |
| 83 | \( 1 + (2.80 - 5.81i)T + (-51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-0.413 - 0.657i)T + (-38.6 + 80.1i)T^{2} \) |
| 97 | \( 1 + (4.94 + 14.1i)T + (-75.8 + 60.4i)T^{2} \) |
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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
−13.54897105240702755827778056251, −12.12126322654371113738005048950, −11.93400517731330619780032493881, −10.52205161678173731320624348353, −9.687237451321105153404101067019, −7.51620686796036543220201230019, −6.34220532617028440921584064416, −5.66402597561515115966080421424, −4.22084861520049996225744187664, −2.14702158400354170867470224027,
2.67958847678011542519347442548, 4.72836840875240430798676594390, 5.53748336621064386208361204835, 6.46019518452345450138620372323, 7.966160863941464396211038673770, 9.738139071618366738980793988251, 10.74560336383396969307164667091, 11.72081281625519173311604768860, 12.60730685048462091907426439386, 13.60984573657865217507121987734
