| L(s) = 1 | + (0.927 − 1.79i)2-s + (−0.342 + 1.69i)3-s + (−1.21 − 1.70i)4-s + (−0.509 + 2.09i)5-s + (2.73 + 2.19i)6-s + (4.00 + 1.38i)7-s + (−0.191 + 0.0275i)8-s + (−2.76 − 1.16i)9-s + (3.30 + 2.86i)10-s + (−0.142 + 2.98i)11-s + (3.31 − 1.47i)12-s + (−2.24 − 6.49i)13-s + (6.21 − 5.92i)14-s + (−3.38 − 1.58i)15-s + (1.24 − 3.59i)16-s + (0.615 + 1.34i)17-s + ⋯ |
| L(s) = 1 | + (0.655 − 1.27i)2-s + (−0.197 + 0.980i)3-s + (−0.607 − 0.853i)4-s + (−0.227 + 0.938i)5-s + (1.11 + 0.894i)6-s + (1.51 + 0.524i)7-s + (−0.0676 + 0.00973i)8-s + (−0.921 − 0.387i)9-s + (1.04 + 0.904i)10-s + (−0.0429 + 0.901i)11-s + (0.956 − 0.427i)12-s + (−0.623 − 1.80i)13-s + (1.66 − 1.58i)14-s + (−0.875 − 0.408i)15-s + (0.310 − 0.897i)16-s + (0.149 + 0.326i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63499 - 0.211801i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63499 - 0.211801i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.342 - 1.69i)T \) |
| 23 | \( 1 + (-3.75 - 2.98i)T \) |
| good | 2 | \( 1 + (-0.927 + 1.79i)T + (-1.16 - 1.62i)T^{2} \) |
| 5 | \( 1 + (0.509 - 2.09i)T + (-4.44 - 2.29i)T^{2} \) |
| 7 | \( 1 + (-4.00 - 1.38i)T + (5.50 + 4.32i)T^{2} \) |
| 11 | \( 1 + (0.142 - 2.98i)T + (-10.9 - 1.04i)T^{2} \) |
| 13 | \( 1 + (2.24 + 6.49i)T + (-10.2 + 8.03i)T^{2} \) |
| 17 | \( 1 + (-0.615 - 1.34i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (5.21 + 2.37i)T + (12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.707 - 0.503i)T + (9.48 + 27.4i)T^{2} \) |
| 31 | \( 1 + (1.63 + 0.653i)T + (22.4 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-0.388 + 1.32i)T + (-31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (11.4 + 2.76i)T + (36.4 + 18.7i)T^{2} \) |
| 43 | \( 1 + (1.78 + 4.46i)T + (-31.1 + 29.6i)T^{2} \) |
| 47 | \( 1 + (-4.19 + 2.42i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.97 + 5.74i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (0.0578 - 0.0200i)T + (46.3 - 36.4i)T^{2} \) |
| 61 | \( 1 + (-4.21 - 5.36i)T + (-14.3 + 59.2i)T^{2} \) |
| 67 | \( 1 + (-1.08 + 0.0516i)T + (66.6 - 6.36i)T^{2} \) |
| 71 | \( 1 + (2.81 + 4.37i)T + (-29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (5.08 - 11.1i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.830 + 4.31i)T + (-73.3 - 29.3i)T^{2} \) |
| 83 | \( 1 + (1.74 + 7.19i)T + (-73.7 + 38.0i)T^{2} \) |
| 89 | \( 1 + (0.911 - 6.33i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-3.28 + 3.44i)T + (-4.61 - 96.8i)T^{2} \) |
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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
−12.07820487765877440062210013730, −11.28373707388761258034291977530, −10.62134975192608558969958027557, −10.09852855825486035167363158016, −8.574701813223213328633983023781, −7.35607133524096078393634259837, −5.36846907165503328046584266629, −4.73085446245529996780459372799, −3.41009595423037403679914780387, −2.30022283023328136733407261842,
1.58810492273686273002124949159, 4.44578742360432478562458789775, 5.03002684655027479479073861490, 6.34038232230312474774084444301, 7.22320400054064582495602282360, 8.217517788189884451004253118569, 8.703508361974623255107229138894, 10.88196403033292848699411990728, 11.71213757682720353697137867648, 12.66999514986206753624779421544
