| L(s) = 1 | + (−0.806 − 1.67i)2-s + (−16.6 + 1.24i)3-s + (37.7 − 47.3i)4-s + (−23.7 − 76.8i)5-s + (15.4 + 26.8i)6-s + (289. + 167. i)7-s + (−225. − 51.5i)8-s + (−445. + 67.2i)9-s + (−109. + 101. i)10-s + (−36.3 − 45.6i)11-s + (−568. + 834. i)12-s + (−2.47e3 − 2.29e3i)13-s + (46.5 − 620. i)14-s + (489. + 1.24e3i)15-s + (−766. − 3.35e3i)16-s + (−7.10e3 − 2.19e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.209i)2-s + (−0.615 + 0.0461i)3-s + (0.589 − 0.739i)4-s + (−0.189 − 0.614i)5-s + (0.0717 + 0.124i)6-s + (0.845 + 0.488i)7-s + (−0.440 − 0.100i)8-s + (−0.611 + 0.0922i)9-s + (−0.109 + 0.101i)10-s + (−0.0273 − 0.0342i)11-s + (−0.329 + 0.482i)12-s + (−1.12 − 1.04i)13-s + (0.0169 − 0.226i)14-s + (0.145 + 0.369i)15-s + (−0.187 − 0.819i)16-s + (−1.44 − 0.446i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.983 + 0.178i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0628490 - 0.697860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0628490 - 0.697860i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.89e3 - 7.92e4i)T \) |
| good | 2 | \( 1 + (0.806 + 1.67i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (16.6 - 1.24i)T + (720. - 108. i)T^{2} \) |
| 5 | \( 1 + (23.7 + 76.8i)T + (-1.29e4 + 8.80e3i)T^{2} \) |
| 7 | \( 1 + (-289. - 167. i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (36.3 + 45.6i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (2.47e3 + 2.29e3i)T + (3.60e5 + 4.81e6i)T^{2} \) |
| 17 | \( 1 + (7.10e3 + 2.19e3i)T + (1.99e7 + 1.35e7i)T^{2} \) |
| 19 | \( 1 + (1.12e3 - 7.48e3i)T + (-4.49e7 - 1.38e7i)T^{2} \) |
| 23 | \( 1 + (3.52e3 - 8.97e3i)T + (-1.08e8 - 1.00e8i)T^{2} \) |
| 29 | \( 1 + (2.45e4 + 1.84e3i)T + (5.88e8 + 8.86e7i)T^{2} \) |
| 31 | \( 1 + (2.05e4 + 1.39e4i)T + (3.24e8 + 8.26e8i)T^{2} \) |
| 37 | \( 1 + (-7.55e4 + 4.36e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-4.59e4 + 2.21e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-1.12e5 + 1.41e5i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (1.57e5 - 1.45e5i)T + (1.65e9 - 2.21e10i)T^{2} \) |
| 59 | \( 1 + (4.63e4 + 2.03e5i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (8.74e4 + 1.28e5i)T + (-1.88e10 + 4.79e10i)T^{2} \) |
| 67 | \( 1 + (-1.46e5 - 2.20e4i)T + (8.64e10 + 2.66e10i)T^{2} \) |
| 71 | \( 1 + (-6.42e4 + 2.52e4i)T + (9.39e10 - 8.71e10i)T^{2} \) |
| 73 | \( 1 + (-2.80e5 + 3.02e5i)T + (-1.13e10 - 1.50e11i)T^{2} \) |
| 79 | \( 1 + (-4.11e4 + 7.12e4i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (3.24e4 + 4.32e5i)T + (-3.23e11 + 4.87e10i)T^{2} \) |
| 89 | \( 1 + (3.36e5 - 2.52e4i)T + (4.91e11 - 7.40e10i)T^{2} \) |
| 97 | \( 1 + (-4.14e5 - 5.19e5i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.42329711660299396227910699506, −12.60468966437182122967932692797, −11.57183719131817745294423141510, −10.79496578097292069204927684262, −9.304369263043982991360542938567, −7.79013539606566645482380617307, −5.90383101815754675038468276289, −4.98250532564890690936397854856, −2.22602257534275319564112355133, −0.33669799728350719996549135407,
2.45290061261028171510398874756, 4.50043709901076601702704911831, 6.50676875664890030574396367410, 7.40305881715421809796159839429, 8.874618058805254032062614738418, 11.07173267420538327453031201849, 11.30095572381026837492326156339, 12.67321969251985600564579903578, 14.31052981067241284648236753466, 15.23288555030530989410226345024
