| L(s) = 1 | + (1.26 + 0.460i)2-s + (−0.347 − 1.96i)3-s + (−0.141 − 0.118i)4-s + (−0.358 + 0.300i)5-s + (0.467 − 2.65i)6-s + (1.17 − 2.03i)7-s + (−1.47 − 2.54i)8-s + (−0.939 + 0.342i)9-s + (−0.592 + 0.215i)10-s + (−0.5 − 0.866i)11-s + (−0.184 + 0.320i)12-s + (−0.819 + 4.64i)13-s + (2.42 − 2.03i)14-s + (0.716 + 0.601i)15-s + (−0.624 − 3.54i)16-s + (4.37 + 1.59i)17-s + ⋯ |
| L(s) = 1 | + (0.895 + 0.325i)2-s + (−0.200 − 1.13i)3-s + (−0.0707 − 0.0593i)4-s + (−0.160 + 0.134i)5-s + (0.191 − 1.08i)6-s + (0.443 − 0.768i)7-s + (−0.520 − 0.901i)8-s + (−0.313 + 0.114i)9-s + (−0.187 + 0.0681i)10-s + (−0.150 − 0.261i)11-s + (−0.0533 + 0.0923i)12-s + (−0.227 + 1.28i)13-s + (0.647 − 0.543i)14-s + (0.185 + 0.155i)15-s + (−0.156 − 0.885i)16-s + (1.06 + 0.385i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33858 - 0.884289i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33858 - 0.884289i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-4.34 + 0.405i)T \) |
| good | 2 | \( 1 + (-1.26 - 0.460i)T + (1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (0.347 + 1.96i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (0.358 - 0.300i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.17 + 2.03i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (0.819 - 4.64i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-4.37 - 1.59i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.315 - 0.264i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.10 + 1.13i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (1.96 - 3.39i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.87T + 37T^{2} \) |
| 41 | \( 1 + (-0.205 - 1.16i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (5.23 - 4.39i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-9.09 + 3.31i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-6.34 - 5.31i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (4.16 + 1.51i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (7.85 + 6.58i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.89 + 1.78i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (6.68 - 5.60i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (2.35 + 13.3i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.103 - 0.587i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.05 + 12.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.51 + 8.61i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-7.73 - 2.81i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.29481075446851616616195784744, −11.71857980679418613612755584051, −10.37764812101926198169738338104, −9.194204967719246384722197048926, −7.64487573389005345057037564491, −7.03332657837918404488271227169, −6.00048155029988305561961940283, −4.81306250970171700443013101784, −3.55891066934152455849093122543, −1.30404639406222892841744389324,
2.82662684781981806462436135561, 3.96178803665317707022645952337, 5.18952814985292116923813858962, 5.48703852180423219489709731960, 7.68411737608993143223102594967, 8.703906579713432385084982907847, 9.825123383807033948243872164951, 10.67521404448103428028153031773, 11.90929177113792407984680984115, 12.30003456222500053221276487841
