| 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
−14.39267951002387973135771019497, −13.06881435182228102329230992828, −12.32075752427596401320797990752, −11.28509184128564086604491695777, −10.56472026597940011508777444694, −8.871219702965045548071490330351, −7.37674187412628349662086740870, −5.07713877443731997798106207211, −4.55971510649005273071579696399, −2.90366515868380419175974481093,
3.42350584161815976167091798682, 5.00263397106456656555672217726, 6.23257347663805426915618412113, 7.44935073740780986947069482989, 8.296009559225838216003765813494, 10.63963524175169434881657186564, 11.63701325631845450852502221712, 12.91151231425418406039699425447, 13.51587157126572107693823290737, 14.81463678640450569929303136338
