| 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
−12.30003456222500053221276487841, −11.90929177113792407984680984115, −10.67521404448103428028153031773, −9.825123383807033948243872164951, −8.703906579713432385084982907847, −7.68411737608993143223102594967, −5.48703852180423219489709731960, −5.18952814985292116923813858962, −3.96178803665317707022645952337, −2.82662684781981806462436135561,
1.30404639406222892841744389324, 3.55891066934152455849093122543, 4.81306250970171700443013101784, 6.00048155029988305561961940283, 7.03332657837918404488271227169, 7.64487573389005345057037564491, 9.194204967719246384722197048926, 10.37764812101926198169738338104, 11.71857980679418613612755584051, 12.29481075446851616616195784744
