| L(s) = 1 | + (−0.446 + 1.34i)2-s + (−1.95 + 0.891i)3-s + (−1.60 − 1.19i)4-s + (−1.03 + 0.895i)5-s + (−0.323 − 3.01i)6-s + (−2.14 − 1.38i)7-s + (2.32 − 1.61i)8-s + (1.04 − 1.21i)9-s + (−0.739 − 1.78i)10-s + (0.745 + 5.18i)11-s + (4.19 + 0.913i)12-s + (−3.68 + 2.37i)13-s + (2.81 − 2.26i)14-s + (1.21 − 2.66i)15-s + (1.12 + 3.83i)16-s + (−0.725 − 2.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.315 + 0.948i)2-s + (−1.12 + 0.514i)3-s + (−0.800 − 0.599i)4-s + (−0.461 + 0.400i)5-s + (−0.132 − 1.23i)6-s + (−0.811 − 0.521i)7-s + (0.821 − 0.570i)8-s + (0.349 − 0.403i)9-s + (−0.233 − 0.564i)10-s + (0.224 + 1.56i)11-s + (1.21 + 0.263i)12-s + (−1.02 + 0.657i)13-s + (0.751 − 0.605i)14-s + (0.314 − 0.688i)15-s + (0.281 + 0.959i)16-s + (−0.175 − 0.598i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0369143 - 0.304636i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0369143 - 0.304636i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.446 - 1.34i)T \) |
| 23 | \( 1 + (-1.53 - 4.54i)T \) |
| good | 3 | \( 1 + (1.95 - 0.891i)T + (1.96 - 2.26i)T^{2} \) |
| 5 | \( 1 + (1.03 - 0.895i)T + (0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (2.14 + 1.38i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.745 - 5.18i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.68 - 2.37i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.725 + 2.46i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-2.83 - 0.832i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-4.83 + 1.42i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (5.16 + 2.35i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.714 + 0.618i)T + (5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.41 - 3.94i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (2.17 + 4.75i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 6.04iT - 47T^{2} \) |
| 53 | \( 1 + (6.78 - 10.5i)T + (-22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-2.90 - 4.51i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (1.40 + 0.643i)T + (39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (2.03 - 14.1i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-6.01 - 0.864i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-13.9 - 4.10i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (1.15 - 0.739i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (6.21 - 7.17i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (6.34 - 2.89i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.24 + 2.81i)T + (13.8 - 96.0i)T^{2} \) |
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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
−15.00382297637725352668586775813, −13.82219633166900034155047133663, −12.43205007526003862208412533872, −11.32643548498073658527904758147, −10.00025427845120007345210191382, −9.530710209496395301889516810847, −7.38154138160215805815809586679, −6.85603782178984402766561245009, −5.32372606440200284184067988817, −4.23001873226722984842228268678,
0.46773010779926880147682104483, 3.16481878844097437489565146392, 5.09372875039499603534421125842, 6.39297946906820040788191254069, 8.084712540872461776231450141165, 9.188462661357770457656466855396, 10.57700772153545217232424840275, 11.45878841424288552045064960830, 12.43796440477046271988823503998, 12.76944856435903707617310525259
