| L(s) = 1 | + (−1.08 + 4.77i)2-s + (−6.48 + 6.01i)3-s + (−14.3 − 6.92i)4-s + (20.4 + 3.07i)5-s + (−21.6 − 37.4i)6-s + (−5.26 + 9.12i)7-s + (24.3 − 30.5i)8-s + (3.82 − 51.0i)9-s + (−36.9 + 94.0i)10-s + (−7.67 + 3.69i)11-s + (134. − 41.6i)12-s + (4.91 + 12.5i)13-s + (−37.8 − 35.0i)14-s + (−150. + 102. i)15-s + (39.4 + 49.4i)16-s + (−46.0 + 6.93i)17-s + ⋯ |
| L(s) = 1 | + (−0.385 + 1.68i)2-s + (−1.24 + 1.15i)3-s + (−1.79 − 0.866i)4-s + (1.82 + 0.275i)5-s + (−1.47 − 2.55i)6-s + (−0.284 + 0.492i)7-s + (1.07 − 1.34i)8-s + (0.141 − 1.88i)9-s + (−1.16 + 2.97i)10-s + (−0.210 + 0.101i)11-s + (3.24 − 1.00i)12-s + (0.104 + 0.266i)13-s + (−0.721 − 0.669i)14-s + (−2.59 + 1.76i)15-s + (0.616 + 0.772i)16-s + (−0.656 + 0.0989i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.674 + 0.738i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.306589 - 0.695014i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.306589 - 0.695014i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-184. - 213. i)T \) |
| good | 2 | \( 1 + (1.08 - 4.77i)T + (-7.20 - 3.47i)T^{2} \) |
| 3 | \( 1 + (6.48 - 6.01i)T + (2.01 - 26.9i)T^{2} \) |
| 5 | \( 1 + (-20.4 - 3.07i)T + (119. + 36.8i)T^{2} \) |
| 7 | \( 1 + (5.26 - 9.12i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (7.67 - 3.69i)T + (829. - 1.04e3i)T^{2} \) |
| 13 | \( 1 + (-4.91 - 12.5i)T + (-1.61e3 + 1.49e3i)T^{2} \) |
| 17 | \( 1 + (46.0 - 6.93i)T + (4.69e3 - 1.44e3i)T^{2} \) |
| 19 | \( 1 + (-1.12 - 15.0i)T + (-6.78e3 + 1.02e3i)T^{2} \) |
| 23 | \( 1 + (-125. - 85.2i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (36.7 + 34.1i)T + (1.82e3 + 2.43e4i)T^{2} \) |
| 31 | \( 1 + (114. - 35.2i)T + (2.46e4 - 1.67e4i)T^{2} \) |
| 37 | \( 1 + (77.8 + 134. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (56.9 - 249. i)T + (-6.20e4 - 2.99e4i)T^{2} \) |
| 47 | \( 1 + (42.2 + 20.3i)T + (6.47e4 + 8.11e4i)T^{2} \) |
| 53 | \( 1 + (-27.1 + 69.2i)T + (-1.09e5 - 1.01e5i)T^{2} \) |
| 59 | \( 1 + (531. + 666. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (-878. - 271. i)T + (1.87e5 + 1.27e5i)T^{2} \) |
| 67 | \( 1 + (-2.74 - 36.6i)T + (-2.97e5 + 4.48e4i)T^{2} \) |
| 71 | \( 1 + (-670. + 457. i)T + (1.30e5 - 3.33e5i)T^{2} \) |
| 73 | \( 1 + (-235. - 599. i)T + (-2.85e5 + 2.64e5i)T^{2} \) |
| 79 | \( 1 + (-254. + 440. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (37.4 - 34.7i)T + (4.27e4 - 5.70e5i)T^{2} \) |
| 89 | \( 1 + (-652. + 605. i)T + (5.26e4 - 7.02e5i)T^{2} \) |
| 97 | \( 1 + (709. - 341. i)T + (5.69e5 - 7.13e5i)T^{2} \) |
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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
−16.31555445706504831536521026692, −15.39570837168968928708208350446, −14.40800054177202433187480018892, −13.08708208309529932138738296709, −10.94513014911417207263239277317, −9.704490022558182482064706324555, −9.159838570423813430864912616183, −6.67733630717904803789231172337, −5.82681572923730607438653567830, −5.05091426795508135376842071657,
0.843970427865145928991548858245, 2.20422453433747175016991553074, 5.25996406226175836335247746541, 6.71229609190411438034549096338, 8.995661363303345524786327423528, 10.32896107569767047355940694098, 11.00152692737104153407276937634, 12.42989702826198105942289310077, 13.12507494295808277174233192571, 13.70290839954813954883544806617
