| L(s) = 1 | + (−0.0796 − 0.0598i)2-s + (−0.377 − 1.85i)3-s + (−0.553 − 1.91i)4-s + (−2.61 + 1.37i)5-s + (−0.0806 + 0.170i)6-s + (−1.55 + 0.252i)7-s + (−0.141 + 0.371i)8-s + (−0.520 + 0.221i)9-s + (0.290 + 0.0472i)10-s + (−0.353 − 0.150i)11-s + (−3.32 + 1.74i)12-s + (−3.28 − 1.49i)13-s + (0.138 + 0.0729i)14-s + (3.53 + 4.32i)15-s + (−3.33 + 2.10i)16-s + (7.06 − 1.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.0563 − 0.0423i)2-s + (−0.218 − 1.06i)3-s + (−0.276 − 0.955i)4-s + (−1.17 + 0.614i)5-s + (−0.0329 + 0.0694i)6-s + (−0.587 + 0.0954i)7-s + (−0.0498 + 0.131i)8-s + (−0.173 + 0.0739i)9-s + (0.0920 + 0.0149i)10-s + (−0.106 − 0.0453i)11-s + (−0.960 + 0.504i)12-s + (−0.909 − 0.414i)13-s + (0.0371 + 0.0194i)14-s + (0.911 + 1.11i)15-s + (−0.832 + 0.526i)16-s + (1.71 − 0.278i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.100583 - 0.553826i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.100583 - 0.553826i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (3.28 + 1.49i)T \) |
| good | 2 | \( 1 + (0.0796 + 0.0598i)T + (0.556 + 1.92i)T^{2} \) |
| 3 | \( 1 + (0.377 + 1.85i)T + (-2.75 + 1.17i)T^{2} \) |
| 5 | \( 1 + (2.61 - 1.37i)T + (2.84 - 4.11i)T^{2} \) |
| 7 | \( 1 + (1.55 - 0.252i)T + (6.63 - 2.21i)T^{2} \) |
| 11 | \( 1 + (0.353 + 0.150i)T + (7.61 + 7.93i)T^{2} \) |
| 17 | \( 1 + (-7.06 + 1.14i)T + (16.1 - 5.38i)T^{2} \) |
| 19 | \( 1 + (3.51 + 6.08i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.65 + 6.33i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.86 + 1.40i)T + (8.06 + 27.8i)T^{2} \) |
| 31 | \( 1 + (-0.855 - 1.23i)T + (-10.9 + 28.9i)T^{2} \) |
| 37 | \( 1 + (-1.79 + 0.144i)T + (36.5 - 5.93i)T^{2} \) |
| 41 | \( 1 + (-1.60 - 7.85i)T + (-37.7 + 16.0i)T^{2} \) |
| 43 | \( 1 + (-10.5 - 0.848i)T + (42.4 + 6.89i)T^{2} \) |
| 47 | \( 1 + (3.87 - 0.954i)T + (41.6 - 21.8i)T^{2} \) |
| 53 | \( 1 + (0.313 - 0.827i)T + (-39.6 - 35.1i)T^{2} \) |
| 59 | \( 1 + (-1.12 - 0.710i)T + (25.2 + 53.3i)T^{2} \) |
| 61 | \( 1 + (4.56 - 5.59i)T + (-12.2 - 59.7i)T^{2} \) |
| 67 | \( 1 + (-2.82 + 9.76i)T + (-56.6 - 35.8i)T^{2} \) |
| 71 | \( 1 + (-8.71 - 2.90i)T + (56.7 + 42.6i)T^{2} \) |
| 73 | \( 1 + (1.20 + 9.94i)T + (-70.8 + 17.4i)T^{2} \) |
| 79 | \( 1 + (2.10 - 0.519i)T + (69.9 - 36.7i)T^{2} \) |
| 83 | \( 1 + (6.93 + 6.14i)T + (10.0 + 82.3i)T^{2} \) |
| 89 | \( 1 + (-6.02 + 10.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.208 + 5.18i)T + (-96.6 + 7.80i)T^{2} \) |
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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
−12.42609495741464274678130328413, −11.41195470373117719997378272633, −10.46532680991392816547734415175, −9.395883067828044464077213993910, −7.909479730929901721716439228019, −7.06465848491166035036244619470, −6.16220110209395532215160800067, −4.66400699700605213892434991237, −2.81367014012435230604878974692, −0.55676034635620001634252638811,
3.53029883493430079420819937406, 4.12193448773480541027411213300, 5.33596141019594222427089029733, 7.34108881759590448090346994414, 8.050729416730773925485646899254, 9.281041773159006055780835622748, 10.05031848708179190772122734455, 11.36127151117072134826018961152, 12.35222468161349197374785112117, 12.75659526265811049985435683465
