| L(s) = 1 | + 6.20i·2-s + (−29.5 + 17.0i)3-s + 25.5·4-s + (−46.5 + 26.8i)5-s + (−105. − 183. i)6-s + (−448. − 259. i)7-s + 555. i·8-s + (218. − 378. i)9-s + (−166. − 288. i)10-s + 668.·11-s + (−754. + 435. i)12-s + (1.45e3 − 2.52e3i)13-s + (1.60e3 − 2.78e3i)14-s + (917. − 1.58e3i)15-s − 1.81e3·16-s + (4.62e3 − 8.00e3i)17-s + ⋯ |
| L(s) = 1 | + 0.775i·2-s + (−1.09 + 0.632i)3-s + 0.398·4-s + (−0.372 + 0.215i)5-s + (−0.490 − 0.849i)6-s + (−1.30 − 0.755i)7-s + 1.08i·8-s + (0.299 − 0.519i)9-s + (−0.166 − 0.288i)10-s + 0.502·11-s + (−0.436 + 0.252i)12-s + (0.664 − 1.15i)13-s + (0.585 − 1.01i)14-s + (0.271 − 0.471i)15-s − 0.442·16-s + (0.940 − 1.62i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.216780 - 0.150458i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.216780 - 0.150458i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (7.66e4 - 2.12e4i)T \) |
| good | 2 | \( 1 - 6.20iT - 64T^{2} \) |
| 3 | \( 1 + (29.5 - 17.0i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (46.5 - 26.8i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (448. + 259. i)T + (5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 - 668.T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-1.45e3 + 2.52e3i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + (-4.62e3 + 8.00e3i)T + (-1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (8.61e3 - 4.97e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-1.26e3 - 2.18e3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (2.89e4 + 1.67e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.21e4 + 2.11e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-2.02e4 + 1.16e4i)T + (1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + 1.15e5T + 4.75e9T^{2} \) |
| 47 | \( 1 - 6.17e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + (7.85e4 + 1.36e5i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + 1.24e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + (-1.42e5 - 8.22e4i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.53e5 + 2.66e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-1.44e5 - 8.36e4i)T + (6.40e10 + 1.10e11i)T^{2} \) |
| 73 | \( 1 + (9.21e4 + 5.32e4i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (4.26e4 - 7.38e4i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-2.15e5 - 3.73e5i)T + (-1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-3.36e5 + 1.94e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.25e6T + 8.32e11T^{2} \) |
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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
−14.95869449976524323442637853324, −13.35974299121166828554029756482, −11.82839852610548466449393669243, −10.86040163420397750025396287986, −9.811488548174890189829300376740, −7.71775918588168451124942321848, −6.49322617002465495418837983780, −5.50302131151754846509552642770, −3.54987458622633281369572807999, −0.13848413660799481942449066795,
1.62527753598556205534325938878, 3.65468102228167398611673199722, 6.12415058937497287020714225275, 6.71649440468010969477700641205, 8.943036733313671440407000599440, 10.47403147427232446925352799660, 11.59999412229275568892861776564, 12.36515578804498137261559582158, 12.98058230055416195283117022160, 15.11598047362730338322164601825
