| L(s) = 1 | + (−0.468 − 0.972i)2-s + (−1.93 + 4.01i)3-s + (1.76 − 2.21i)4-s + (8.30 + 1.89i)5-s + 4.81·6-s + 6.84i·7-s + (−7.19 − 1.64i)8-s + (−6.77 − 8.49i)9-s + (−2.04 − 8.96i)10-s + (−8.76 − 10.9i)11-s + (5.48 + 11.3i)12-s + (1.36 − 5.99i)13-s + (6.65 − 3.20i)14-s + (−23.6 + 29.6i)15-s + (−0.749 − 3.28i)16-s + (−0.615 − 2.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.234 − 0.486i)2-s + (−0.644 + 1.33i)3-s + (0.441 − 0.553i)4-s + (1.66 + 0.379i)5-s + 0.802·6-s + 0.977i·7-s + (−0.899 − 0.205i)8-s + (−0.752 − 0.943i)9-s + (−0.204 − 0.896i)10-s + (−0.796 − 0.998i)11-s + (0.456 + 0.948i)12-s + (0.105 − 0.461i)13-s + (0.475 − 0.229i)14-s + (−1.57 + 1.97i)15-s + (−0.0468 − 0.205i)16-s + (−0.0361 − 0.158i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.02140 + 0.178646i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.02140 + 0.178646i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-27.5 + 32.9i)T \) |
| good | 2 | \( 1 + (0.468 + 0.972i)T + (-2.49 + 3.12i)T^{2} \) |
| 3 | \( 1 + (1.93 - 4.01i)T + (-5.61 - 7.03i)T^{2} \) |
| 5 | \( 1 + (-8.30 - 1.89i)T + (22.5 + 10.8i)T^{2} \) |
| 7 | \( 1 - 6.84iT - 49T^{2} \) |
| 11 | \( 1 + (8.76 + 10.9i)T + (-26.9 + 117. i)T^{2} \) |
| 13 | \( 1 + (-1.36 + 5.99i)T + (-152. - 73.3i)T^{2} \) |
| 17 | \( 1 + (0.615 + 2.69i)T + (-260. + 125. i)T^{2} \) |
| 19 | \( 1 + (-10.2 - 8.20i)T + (80.3 + 351. i)T^{2} \) |
| 23 | \( 1 + (11.5 + 14.4i)T + (-117. + 515. i)T^{2} \) |
| 29 | \( 1 + (22.3 + 46.4i)T + (-524. + 657. i)T^{2} \) |
| 31 | \( 1 + (10.3 - 4.98i)T + (599. - 751. i)T^{2} \) |
| 37 | \( 1 - 53.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-7.30 + 3.51i)T + (1.04e3 - 1.31e3i)T^{2} \) |
| 47 | \( 1 + (-18.4 + 23.1i)T + (-491. - 2.15e3i)T^{2} \) |
| 53 | \( 1 + (-7.84 - 34.3i)T + (-2.53e3 + 1.21e3i)T^{2} \) |
| 59 | \( 1 + (-23.1 - 101. i)T + (-3.13e3 + 1.51e3i)T^{2} \) |
| 61 | \( 1 + (33.6 - 69.8i)T + (-2.32e3 - 2.90e3i)T^{2} \) |
| 67 | \( 1 + (-1.19 + 1.50i)T + (-998. - 4.37e3i)T^{2} \) |
| 71 | \( 1 + (-6.33 - 5.04i)T + (1.12e3 + 4.91e3i)T^{2} \) |
| 73 | \( 1 + (59.4 + 13.5i)T + (4.80e3 + 2.31e3i)T^{2} \) |
| 79 | \( 1 - 22.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + (37.5 + 18.0i)T + (4.29e3 + 5.38e3i)T^{2} \) |
| 89 | \( 1 + (7.92 - 16.4i)T + (-4.93e3 - 6.19e3i)T^{2} \) |
| 97 | \( 1 + (40.8 + 51.2i)T + (-2.09e3 + 9.17e3i)T^{2} \) |
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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
−15.76102604921041929642125758887, −14.92653660411053384933095610108, −13.57961329126888253896685652376, −11.78349379435204450385464352392, −10.59854933443485939327455308619, −10.08269549639160372819203714416, −9.073935814079184421932554344306, −5.85325323788850399697011471660, −5.62659583927962498344181956486, −2.64701289545632065823527844308,
1.90194890860584164705421413696, 5.56739591002581868792388297704, 6.79928818948162487709636049062, 7.58313851369740876408873120340, 9.429762477537192802251714083038, 10.98196358998510663211103186936, 12.56418165544580438737984922896, 13.09686235531840888810529529562, 14.21069565102161167661250296775, 16.15968740902908168590319140176
