| L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (3.75 − 1.36i)5-s + (−0.766 + 0.642i)6-s + (−1.5 − 2.59i)7-s + (0.5 − 0.866i)8-s + (−0.347 − 1.96i)9-s + (0.694 + 3.93i)10-s + (−1 + 1.73i)11-s + (−0.499 − 0.866i)12-s + (0.766 − 0.642i)13-s + (2.81 − 1.02i)14-s + (3.75 + 1.36i)15-s + (0.766 + 0.642i)16-s + (0.520 − 2.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.442 + 0.371i)3-s + (−0.469 − 0.171i)4-s + (1.68 − 0.611i)5-s + (−0.312 + 0.262i)6-s + (−0.566 − 0.981i)7-s + (0.176 − 0.306i)8-s + (−0.115 − 0.656i)9-s + (0.219 + 1.24i)10-s + (−0.301 + 0.522i)11-s + (−0.144 − 0.249i)12-s + (0.212 − 0.178i)13-s + (0.753 − 0.274i)14-s + (0.970 + 0.353i)15-s + (0.191 + 0.160i)16-s + (0.126 − 0.716i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.90907 - 0.00124606i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90907 - 0.00124606i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (-0.766 - 0.642i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (-3.75 + 1.36i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.766 + 0.642i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.520 + 2.95i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.868 - 4.92i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (-6.12 - 5.14i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.75 - 1.36i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.38 - 7.87i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (2.60 - 14.7i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (1.87 + 0.684i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.520 + 2.95i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.87 + 0.684i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-6.89 - 5.78i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.66 - 6.42i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.347 + 1.96i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−9.946385458866876128409090674017, −9.505712989964111900621959079436, −8.993701741030379317684023459054, −7.79434317503600615902412098691, −6.77353212831371083229918617664, −6.03628928960551209776478725869, −5.11038837896771099169192459280, −4.09033629433171989046159045923, −2.73782565787091805092682040866, −1.05577232352212990458060962920,
1.80585112243372026663086147301, 2.47006291817472578480119666090, 3.33047867291009075337380405714, 5.20177896065939709604276542391, 5.88854829165250777241906922117, 6.75815530400860985243215398441, 8.086342359142549424141074073998, 8.942395444038279752766556746361, 9.523242667902387828733021450479, 10.49156269761228305893753983231
