| L(s) = 1 | + (0.00255 − 0.00280i)2-s + (1.02 − 1.83i)3-s + (0.184 + 1.99i)4-s + (−0.387 + 2.77i)5-s + (−0.00253 − 0.00755i)6-s + (0.121 − 0.515i)7-s + (0.0121 + 0.00914i)8-s + (−0.743 − 1.20i)9-s + (0.00680 + 0.00819i)10-s + (3.29 + 2.73i)11-s + (3.84 + 1.69i)12-s + (−0.720 + 0.953i)13-s + (−0.00113 − 0.00165i)14-s + (4.70 + 3.55i)15-s + (−3.93 + 0.734i)16-s + (−1.08 − 3.97i)17-s + ⋯ |
| L(s) = 1 | + (0.00180 − 0.00198i)2-s + (0.590 − 1.05i)3-s + (0.0922 + 0.995i)4-s + (−0.173 + 1.24i)5-s + (−0.00103 − 0.00308i)6-s + (0.0457 − 0.194i)7-s + (0.00428 + 0.00323i)8-s + (−0.247 − 0.400i)9-s + (0.00215 + 0.00259i)10-s + (0.992 + 0.823i)11-s + (1.10 + 0.489i)12-s + (−0.199 + 0.264i)13-s + (−0.000303 − 0.000442i)14-s + (1.21 + 0.917i)15-s + (−0.982 + 0.183i)16-s + (−0.263 − 0.964i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.51294 + 0.395999i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51294 + 0.395999i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (1.08 + 3.97i)T \) |
| good | 2 | \( 1 + (-0.00255 + 0.00280i)T + (-0.184 - 1.99i)T^{2} \) |
| 3 | \( 1 + (-1.02 + 1.83i)T + (-1.57 - 2.55i)T^{2} \) |
| 5 | \( 1 + (0.387 - 2.77i)T + (-4.80 - 1.36i)T^{2} \) |
| 7 | \( 1 + (-0.121 + 0.515i)T + (-6.26 - 3.12i)T^{2} \) |
| 11 | \( 1 + (-3.29 - 2.73i)T + (2.02 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.720 - 0.953i)T + (-3.55 - 12.5i)T^{2} \) |
| 19 | \( 1 + (-1.13 - 1.23i)T + (-1.75 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-0.827 + 3.51i)T + (-20.5 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-5.48 + 6.61i)T + (-5.32 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.81 - 0.531i)T + (29.8 - 8.48i)T^{2} \) |
| 37 | \( 1 + (1.93 + 4.38i)T + (-24.9 + 27.3i)T^{2} \) |
| 41 | \( 1 + (4.97 + 2.77i)T + (21.5 + 34.8i)T^{2} \) |
| 43 | \( 1 + (1.13 - 6.04i)T + (-40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + (7.76 + 4.80i)T + (20.9 + 42.0i)T^{2} \) |
| 53 | \( 1 + (2.32 + 3.75i)T + (-23.6 + 47.4i)T^{2} \) |
| 59 | \( 1 + (4.62 - 2.30i)T + (35.5 - 47.0i)T^{2} \) |
| 61 | \( 1 + (-3.06 - 9.13i)T + (-48.6 + 36.7i)T^{2} \) |
| 67 | \( 1 + (-9.90 + 9.02i)T + (6.18 - 66.7i)T^{2} \) |
| 71 | \( 1 + (7.36 + 1.73i)T + (63.5 + 31.6i)T^{2} \) |
| 73 | \( 1 + (-1.45 - 0.997i)T + (26.3 + 68.0i)T^{2} \) |
| 79 | \( 1 + (0.555 - 12.0i)T + (-78.6 - 7.28i)T^{2} \) |
| 83 | \( 1 + (2.71 + 0.773i)T + (70.5 + 43.6i)T^{2} \) |
| 89 | \( 1 + (7.72 + 10.2i)T + (-24.3 + 85.6i)T^{2} \) |
| 97 | \( 1 + (4.89 + 1.15i)T + (86.8 + 43.2i)T^{2} \) |
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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
−11.98484342822338305202994689323, −11.24484850937952331311035713475, −9.962225296346493123788501448801, −8.779998085922885320667651114745, −7.75679062432672610416101245742, −7.00548429641629284393084106003, −6.67263360582527085205564739468, −4.37989500689356979027726864044, −3.10985063402053816791562703418, −2.14009386832941865780214782108,
1.31855888988502125432394144249, 3.44474621640680260631787495725, 4.60650261449293147838077491737, 5.39632599000543739334358777339, 6.62592151717170582352421294993, 8.459375827083402128711980327574, 8.936836058664880782702398476256, 9.694005360535859998489057945311, 10.61247556430642892475801469853, 11.58789132416455927967443067453
