| L(s) = 1 | + (6.66 + 13.8i)2-s + (−20.8 − 30.5i)3-s + (−107. + 134. i)4-s + (161. − 173. i)5-s + (283. − 491. i)6-s + (325. − 188. i)7-s + (−1.61e3 − 369. i)8-s + (−232. + 592. i)9-s + (3.47e3 + 1.07e3i)10-s + (54.5 + 68.4i)11-s + (6.34e3 + 475. i)12-s + (−439. + 135. i)13-s + (4.77e3 + 3.25e3i)14-s + (−8.65e3 − 1.30e3i)15-s + (−3.22e3 − 1.41e4i)16-s + (2.67e3 − 2.47e3i)17-s + ⋯ |
| L(s) = 1 | + (0.833 + 1.73i)2-s + (−0.771 − 1.13i)3-s + (−1.67 + 2.10i)4-s + (1.28 − 1.38i)5-s + (1.31 − 2.27i)6-s + (0.950 − 0.548i)7-s + (−3.16 − 0.721i)8-s + (−0.319 + 0.813i)9-s + (3.47 + 1.07i)10-s + (0.0410 + 0.0514i)11-s + (3.66 + 0.274i)12-s + (−0.200 + 0.0617i)13-s + (1.74 + 1.18i)14-s + (−2.56 − 0.386i)15-s + (−0.787 − 3.44i)16-s + (0.543 − 0.504i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.953 - 0.301i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.28965 + 0.353702i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28965 + 0.353702i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (7.09e4 - 3.57e4i)T \) |
| good | 2 | \( 1 + (-6.66 - 13.8i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (20.8 + 30.5i)T + (-266. + 678. i)T^{2} \) |
| 5 | \( 1 + (-161. + 173. i)T + (-1.16e3 - 1.55e4i)T^{2} \) |
| 7 | \( 1 + (-325. + 188. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-54.5 - 68.4i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (439. - 135. i)T + (3.98e6 - 2.71e6i)T^{2} \) |
| 17 | \( 1 + (-2.67e3 + 2.47e3i)T + (1.80e6 - 2.40e7i)T^{2} \) |
| 19 | \( 1 + (-6.89e3 + 2.70e3i)T + (3.44e7 - 3.19e7i)T^{2} \) |
| 23 | \( 1 + (-9.89e3 + 1.49e3i)T + (1.41e8 - 4.36e7i)T^{2} \) |
| 29 | \( 1 + (8.39e3 - 1.23e4i)T + (-2.17e8 - 5.53e8i)T^{2} \) |
| 31 | \( 1 + (832. - 1.11e4i)T + (-8.77e8 - 1.32e8i)T^{2} \) |
| 37 | \( 1 + (9.43e3 + 5.44e3i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (-9.30e3 + 4.48e3i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (2.37e4 - 2.97e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (3.51e4 + 1.08e4i)T + (1.83e10 + 1.24e10i)T^{2} \) |
| 59 | \( 1 + (-3.17e4 - 1.38e5i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-1.46e5 + 1.09e4i)T + (5.09e10 - 7.67e9i)T^{2} \) |
| 67 | \( 1 + (1.22e4 + 3.10e4i)T + (-6.63e10 + 6.15e10i)T^{2} \) |
| 71 | \( 1 + (8.81e4 - 5.85e5i)T + (-1.22e11 - 3.77e10i)T^{2} \) |
| 73 | \( 1 + (1.80e5 + 5.84e5i)T + (-1.25e11 + 8.52e10i)T^{2} \) |
| 79 | \( 1 + (-4.71e5 - 8.16e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-3.37e5 + 2.30e5i)T + (1.19e11 - 3.04e11i)T^{2} \) |
| 89 | \( 1 + (-2.56e5 - 3.76e5i)T + (-1.81e11 + 4.62e11i)T^{2} \) |
| 97 | \( 1 + (-4.06e5 - 5.09e5i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.44426270813704237368732561416, −13.58190755075141867637378874951, −12.92361147431431441452943152608, −11.94429629939640289317567363441, −9.230025428756277754939291391023, −7.898968448387474951260902128046, −6.79835913804382959612887667963, −5.49394699587520893906183501157, −4.90210089660694169193491920285, −1.04762697634008022963832366873,
1.88078041869186451536730278668, 3.34482789215564945435630537912, 5.08395309661552189518621195675, 5.83384459176297974695867818539, 9.513642336533303111530674938281, 10.22903081209865016356054991523, 11.03094981004156993089788367348, 11.77252650935288307117781167099, 13.43805065049344586556355077883, 14.51498893710658354903427581077
