| L(s) = 1 | + (1.63 + 1.18i)2-s + (−0.309 − 0.951i)3-s + (0.641 + 1.97i)4-s + (−2.07 + 0.843i)5-s + (0.623 − 1.92i)6-s + 1.01·7-s + (−0.0473 + 0.145i)8-s + (−0.809 + 0.587i)9-s + (−4.38 − 1.07i)10-s + (−3.85 − 2.79i)11-s + (1.67 − 1.22i)12-s + (0.0840 − 0.0610i)13-s + (1.66 + 1.20i)14-s + (1.44 + 1.70i)15-s + (3.10 − 2.25i)16-s + (−1.80 + 5.55i)17-s + ⋯ |
| L(s) = 1 | + (1.15 + 0.839i)2-s + (−0.178 − 0.549i)3-s + (0.320 + 0.987i)4-s + (−0.926 + 0.377i)5-s + (0.254 − 0.783i)6-s + 0.385·7-s + (−0.0167 + 0.0514i)8-s + (−0.269 + 0.195i)9-s + (−1.38 − 0.341i)10-s + (−1.16 − 0.843i)11-s + (0.484 − 0.352i)12-s + (0.0232 − 0.0169i)13-s + (0.444 + 0.323i)14-s + (0.372 + 0.441i)15-s + (0.777 − 0.564i)16-s + (−0.437 + 1.34i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27078 + 0.426864i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27078 + 0.426864i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.309 + 0.951i)T \) |
| 5 | \( 1 + (2.07 - 0.843i)T \) |
| good | 2 | \( 1 + (-1.63 - 1.18i)T + (0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 + (3.85 + 2.79i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.0840 + 0.0610i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.80 - 5.55i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (0.223 - 0.688i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-7.33 - 5.33i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.23 + 3.79i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.329 + 1.01i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (3.25 - 2.36i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (5.83 - 4.23i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 8.62T + 43T^{2} \) |
| 47 | \( 1 + (2.53 + 7.79i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.34 + 4.15i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-3.97 + 2.88i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (5.63 + 4.09i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (3.06 - 9.43i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (3.33 + 10.2i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (6.98 + 5.07i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.767 - 2.36i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.31 + 4.03i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-14.8 - 10.8i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-2.07 - 6.37i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−14.81463678640450569929303136338, −13.51587157126572107693823290737, −12.91151231425418406039699425447, −11.63701325631845450852502221712, −10.63963524175169434881657186564, −8.296009559225838216003765813494, −7.44935073740780986947069482989, −6.23257347663805426915618412113, −5.00263397106456656555672217726, −3.42350584161815976167091798682,
2.90366515868380419175974481093, 4.55971510649005273071579696399, 5.07713877443731997798106207211, 7.37674187412628349662086740870, 8.871219702965045548071490330351, 10.56472026597940011508777444694, 11.28509184128564086604491695777, 12.32075752427596401320797990752, 13.06881435182228102329230992828, 14.39267951002387973135771019497
