| L(s) = 1 | + (1.68 − 3.49i)2-s + (5.89 + 0.441i)3-s + (30.5 + 38.2i)4-s + (−19.8 + 64.4i)5-s + (11.4 − 19.8i)6-s + (−541. + 312. i)7-s + (427. − 97.4i)8-s + (−686. − 103. i)9-s + (191. + 177. i)10-s + (371. − 465. i)11-s + (163. + 239. i)12-s + (−2.39e3 + 2.22e3i)13-s + (181. + 2.41e3i)14-s + (−145. + 370. i)15-s + (−319. + 1.39e3i)16-s + (6.77e3 − 2.09e3i)17-s + ⋯ |
| L(s) = 1 | + (0.210 − 0.436i)2-s + (0.218 + 0.0163i)3-s + (0.477 + 0.598i)4-s + (−0.158 + 0.515i)5-s + (0.0530 − 0.0918i)6-s + (−1.57 + 0.912i)7-s + (0.834 − 0.190i)8-s + (−0.941 − 0.141i)9-s + (0.191 + 0.177i)10-s + (0.279 − 0.349i)11-s + (0.0943 + 0.138i)12-s + (−1.08 + 1.01i)13-s + (0.0660 + 0.881i)14-s + (−0.0431 + 0.109i)15-s + (−0.0779 + 0.341i)16-s + (1.37 − 0.425i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.880476 + 1.08720i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.880476 + 1.08720i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-7.06e4 + 3.64e4i)T \) |
| good | 2 | \( 1 + (-1.68 + 3.49i)T + (-39.9 - 50.0i)T^{2} \) |
| 3 | \( 1 + (-5.89 - 0.441i)T + (720. + 108. i)T^{2} \) |
| 5 | \( 1 + (19.8 - 64.4i)T + (-1.29e4 - 8.80e3i)T^{2} \) |
| 7 | \( 1 + (541. - 312. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-371. + 465. i)T + (-3.94e5 - 1.72e6i)T^{2} \) |
| 13 | \( 1 + (2.39e3 - 2.22e3i)T + (3.60e5 - 4.81e6i)T^{2} \) |
| 17 | \( 1 + (-6.77e3 + 2.09e3i)T + (1.99e7 - 1.35e7i)T^{2} \) |
| 19 | \( 1 + (-1.26e3 - 8.37e3i)T + (-4.49e7 + 1.38e7i)T^{2} \) |
| 23 | \( 1 + (1.86e3 + 4.74e3i)T + (-1.08e8 + 1.00e8i)T^{2} \) |
| 29 | \( 1 + (-2.53e3 + 190. i)T + (5.88e8 - 8.86e7i)T^{2} \) |
| 31 | \( 1 + (-1.30e4 + 8.89e3i)T + (3.24e8 - 8.26e8i)T^{2} \) |
| 37 | \( 1 + (5.34e3 + 3.08e3i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (1.04e5 + 5.02e4i)T + (2.96e9 + 3.71e9i)T^{2} \) |
| 47 | \( 1 + (-3.99e4 - 5.00e4i)T + (-2.39e9 + 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-8.38e4 - 7.78e4i)T + (1.65e9 + 2.21e10i)T^{2} \) |
| 59 | \( 1 + (6.50e4 - 2.84e5i)T + (-3.80e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (1.78e5 - 2.61e5i)T + (-1.88e10 - 4.79e10i)T^{2} \) |
| 67 | \( 1 + (2.33e5 - 3.51e4i)T + (8.64e10 - 2.66e10i)T^{2} \) |
| 71 | \( 1 + (1.51e4 + 5.94e3i)T + (9.39e10 + 8.71e10i)T^{2} \) |
| 73 | \( 1 + (-1.30e4 - 1.40e4i)T + (-1.13e10 + 1.50e11i)T^{2} \) |
| 79 | \( 1 + (-1.63e5 - 2.83e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-6.74e4 + 9.00e5i)T + (-3.23e11 - 4.87e10i)T^{2} \) |
| 89 | \( 1 + (-1.35e6 - 1.01e5i)T + (4.91e11 + 7.40e10i)T^{2} \) |
| 97 | \( 1 + (-1.26e5 + 1.58e5i)T + (-1.85e11 - 8.12e11i)T^{2} \) |
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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
−14.93301173586876819057927001568, −13.82128982404832810736286351855, −12.16479267585613239591324675162, −11.98703011859425515149405082031, −10.21444416404811305237404343077, −8.958977248390804888220666743437, −7.30552612055732763082429476834, −5.99898792526933501004112322541, −3.46245227513303205223647595818, −2.61651822399659517822947168643,
0.57029526821508396881645369703, 3.06131101098747966343718953937, 5.13287317386291827770184985154, 6.54455358892126698271993490945, 7.74039521926879615503512749618, 9.592044321269230784491401177159, 10.49505072715980398868705967930, 12.18084634961047436860028920625, 13.35330422243579229757347010095, 14.44467358204371032092615094947
