| L(s) = 1 | + (−9.55 + 19.8i)2-s + (33.3 − 48.9i)3-s + (−142. − 179. i)4-s + (−2.19 − 2.37i)5-s + (652. + 1.13e3i)6-s + (65.9 + 38.0i)7-s + (−580. + 132. i)8-s + (1.11e3 + 2.83e3i)9-s + (68.0 − 20.9i)10-s + (6.94e3 − 8.71e3i)11-s + (−1.35e4 + 1.01e3i)12-s + (−5.00e4 − 1.54e4i)13-s + (−1.38e3 + 944. i)14-s + (−189. + 28.5i)15-s + (1.59e4 − 6.99e4i)16-s + (−6.05e4 − 5.61e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.597 + 1.24i)2-s + (0.412 − 0.604i)3-s + (−0.557 − 0.699i)4-s + (−0.00351 − 0.00379i)5-s + (0.503 + 0.872i)6-s + (0.0274 + 0.0158i)7-s + (−0.141 + 0.0323i)8-s + (0.169 + 0.432i)9-s + (0.00680 − 0.00209i)10-s + (0.474 − 0.595i)11-s + (−0.652 + 0.0489i)12-s + (−1.75 − 0.540i)13-s + (−0.0360 + 0.0245i)14-s + (−0.00374 + 0.000564i)15-s + (0.243 − 1.06i)16-s + (−0.724 − 0.672i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.632798 - 0.367421i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.632798 - 0.367421i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.19e6 - 3.20e6i)T \) |
| good | 2 | \( 1 + (9.55 - 19.8i)T + (-159. - 200. i)T^{2} \) |
| 3 | \( 1 + (-33.3 + 48.9i)T + (-2.39e3 - 6.10e3i)T^{2} \) |
| 5 | \( 1 + (2.19 + 2.37i)T + (-2.91e4 + 3.89e5i)T^{2} \) |
| 7 | \( 1 + (-65.9 - 38.0i)T + (2.88e6 + 4.99e6i)T^{2} \) |
| 11 | \( 1 + (-6.94e3 + 8.71e3i)T + (-4.76e7 - 2.08e8i)T^{2} \) |
| 13 | \( 1 + (5.00e4 + 1.54e4i)T + (6.73e8 + 4.59e8i)T^{2} \) |
| 17 | \( 1 + (6.05e4 + 5.61e4i)T + (5.21e8 + 6.95e9i)T^{2} \) |
| 19 | \( 1 + (1.36e3 + 535. i)T + (1.24e10 + 1.15e10i)T^{2} \) |
| 23 | \( 1 + (-2.39e5 - 3.61e4i)T + (7.48e10 + 2.30e10i)T^{2} \) |
| 29 | \( 1 + (1.66e5 + 2.43e5i)T + (-1.82e11 + 4.65e11i)T^{2} \) |
| 31 | \( 1 + (8.44e4 + 1.12e6i)T + (-8.43e11 + 1.27e11i)T^{2} \) |
| 37 | \( 1 + (2.74e4 - 1.58e4i)T + (1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + (3.48e6 + 1.67e6i)T + (4.97e12 + 6.24e12i)T^{2} \) |
| 47 | \( 1 + (4.25e6 + 5.33e6i)T + (-5.29e12 + 2.32e13i)T^{2} \) |
| 53 | \( 1 + (-6.01e6 + 1.85e6i)T + (5.14e13 - 3.50e13i)T^{2} \) |
| 59 | \( 1 + (-1.31e6 + 5.78e6i)T + (-1.32e14 - 6.37e13i)T^{2} \) |
| 61 | \( 1 + (5.33e6 + 3.99e5i)T + (1.89e14 + 2.85e13i)T^{2} \) |
| 67 | \( 1 + (4.09e6 - 1.04e7i)T + (-2.97e14 - 2.76e14i)T^{2} \) |
| 71 | \( 1 + (3.01e6 + 1.99e7i)T + (-6.17e14 + 1.90e14i)T^{2} \) |
| 73 | \( 1 + (1.29e7 - 4.19e7i)T + (-6.66e14 - 4.54e14i)T^{2} \) |
| 79 | \( 1 + (1.86e7 - 3.23e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + (1.66e7 + 1.13e7i)T + (8.22e14 + 2.09e15i)T^{2} \) |
| 89 | \( 1 + (-2.43e7 + 3.57e7i)T + (-1.43e15 - 3.66e15i)T^{2} \) |
| 97 | \( 1 + (5.78e7 - 7.24e7i)T + (-1.74e15 - 7.64e15i)T^{2} \) |
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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
−14.31908616566735011092435594773, −13.12761238403069128524035951165, −11.72625653567887869743331720721, −9.882489311915140634164111517501, −8.612430120110460825159410492443, −7.59605920329768171874352894890, −6.72267460465339185195869500678, −5.09555499393734371737679769541, −2.53943626067743618503147556968, −0.31251594137441597019080191101,
1.62184603830169761476402560988, 3.09509120298049324233792046128, 4.53850941575504482647698991997, 6.92615948990264146424200296670, 8.919606303979909298267539345691, 9.587558913221611248609335065894, 10.59442007625752333007034285784, 11.89871087977826499660092674722, 12.75104469308745769576485362640, 14.61103342091734520366627554618
