| L(s) = 1 | + (5.80 + 7.28i)2-s + (−11.9 − 1.80i)3-s + (−12.1 + 53.4i)4-s + (−56.0 − 38.2i)5-s + (−56.5 − 97.9i)6-s + (−58.6 + 101. i)7-s + (−191. + 92.1i)8-s + (−91.4 − 28.2i)9-s + (−47.2 − 630. i)10-s + (−31.3 − 137. i)11-s + (242. − 619. i)12-s + (−65.8 + 878. i)13-s + (−1.08e3 + 162. i)14-s + (603. + 560. i)15-s + (−203. − 97.9i)16-s + (160. − 109. i)17-s + ⋯ |
| L(s) = 1 | + (1.02 + 1.28i)2-s + (−0.769 − 0.116i)3-s + (−0.381 + 1.66i)4-s + (−1.00 − 0.683i)5-s + (−0.641 − 1.11i)6-s + (−0.452 + 0.783i)7-s + (−1.05 + 0.509i)8-s + (−0.376 − 0.116i)9-s + (−0.149 − 1.99i)10-s + (−0.0780 − 0.341i)11-s + (0.487 − 1.24i)12-s + (−0.108 + 1.44i)13-s + (−1.47 + 0.222i)14-s + (0.692 + 0.642i)15-s + (−0.198 − 0.0956i)16-s + (0.134 − 0.0918i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.593i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.239186 - 0.726699i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.239186 - 0.726699i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-7.85e3 - 9.23e3i)T \) |
| good | 2 | \( 1 + (-5.80 - 7.28i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (11.9 + 1.80i)T + (232. + 71.6i)T^{2} \) |
| 5 | \( 1 + (56.0 + 38.2i)T + (1.14e3 + 2.90e3i)T^{2} \) |
| 7 | \( 1 + (58.6 - 101. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (31.3 + 137. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (65.8 - 878. i)T + (-3.67e5 - 5.53e4i)T^{2} \) |
| 17 | \( 1 + (-160. + 109. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (-529. + 163. i)T + (2.04e6 - 1.39e6i)T^{2} \) |
| 23 | \( 1 + (3.30e3 - 3.06e3i)T + (4.80e5 - 6.41e6i)T^{2} \) |
| 29 | \( 1 + (5.20e3 - 785. i)T + (1.95e7 - 6.04e6i)T^{2} \) |
| 31 | \( 1 + (-1.00e3 + 2.56e3i)T + (-2.09e7 - 1.94e7i)T^{2} \) |
| 37 | \( 1 + (-667. - 1.15e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (-96.5 - 121. i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (-6.08e3 + 2.66e4i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (317. + 4.23e3i)T + (-4.13e8 + 6.23e7i)T^{2} \) |
| 59 | \( 1 + (-7.04e3 - 3.39e3i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (1.27e4 + 3.25e4i)T + (-6.19e8 + 5.74e8i)T^{2} \) |
| 67 | \( 1 + (3.09e4 - 9.54e3i)T + (1.11e9 - 7.60e8i)T^{2} \) |
| 71 | \( 1 + (4.03e4 + 3.74e4i)T + (1.34e8 + 1.79e9i)T^{2} \) |
| 73 | \( 1 + (6.24e3 - 8.33e4i)T + (-2.04e9 - 3.08e8i)T^{2} \) |
| 79 | \( 1 + (-1.84e4 + 3.19e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (3.44e4 + 5.18e3i)T + (3.76e9 + 1.16e9i)T^{2} \) |
| 89 | \( 1 + (-1.28e5 - 1.93e4i)T + (5.33e9 + 1.64e9i)T^{2} \) |
| 97 | \( 1 + (-2.54e4 - 1.11e5i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
| 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
−15.87981238983560451573576317917, −14.65917214086631169160126121201, −13.43368785652860393287790658293, −12.12023032095682467181768278040, −11.66335298650286656938651964506, −9.045269818143855774248178598790, −7.70696390546453064558285105516, −6.29855980569081555164875176160, −5.30465039299324815669979398278, −3.89577436155410705151452178746,
0.32061474046605194823506625236, 3.00162150916667929424484502798, 4.23146365107421159631499898643, 5.79370093898768406953174929116, 7.64001932452325134620338145765, 10.28931765106866459944518576017, 10.78618968990867351327487884806, 11.87091413248575368553575329652, 12.70867418830348629453996732436, 14.01800151122458068987884844703
