| L(s) = 1 | + (1.33 + 5.86i)2-s + (10.9 + 10.2i)3-s + (−3.76 + 1.81i)4-s + (2.77 − 0.417i)5-s + (−45.1 + 78.1i)6-s + (82.5 + 142. i)7-s + (104. + 130. i)8-s + (−1.33 − 17.7i)9-s + (6.15 + 15.6i)10-s + (−493. − 237. i)11-s + (−59.8 − 18.4i)12-s + (51.7 − 131. i)13-s + (−727. + 675. i)14-s + (34.7 + 23.6i)15-s + (−710. + 891. i)16-s + (200. + 30.2i)17-s + ⋯ |
| L(s) = 1 | + (0.236 + 1.03i)2-s + (0.705 + 0.654i)3-s + (−0.117 + 0.0566i)4-s + (0.0495 − 0.00747i)5-s + (−0.511 + 0.886i)6-s + (0.636 + 1.10i)7-s + (0.576 + 0.722i)8-s + (−0.00548 − 0.0732i)9-s + (0.0194 + 0.0496i)10-s + (−1.23 − 0.592i)11-s + (−0.120 − 0.0370i)12-s + (0.0848 − 0.216i)13-s + (−0.992 + 0.920i)14-s + (0.0398 + 0.0271i)15-s + (−0.694 + 0.870i)16-s + (0.168 + 0.0253i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 - 0.868i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.494 - 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.25240 + 2.15471i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.25240 + 2.15471i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (9.75e3 - 7.20e3i)T \) |
| good | 2 | \( 1 + (-1.33 - 5.86i)T + (-28.8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (-10.9 - 10.2i)T + (18.1 + 242. i)T^{2} \) |
| 5 | \( 1 + (-2.77 + 0.417i)T + (2.98e3 - 921. i)T^{2} \) |
| 7 | \( 1 + (-82.5 - 142. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (493. + 237. i)T + (1.00e5 + 1.25e5i)T^{2} \) |
| 13 | \( 1 + (-51.7 + 131. i)T + (-2.72e5 - 2.52e5i)T^{2} \) |
| 17 | \( 1 + (-200. - 30.2i)T + (1.35e6 + 4.18e5i)T^{2} \) |
| 19 | \( 1 + (-19.8 + 265. i)T + (-2.44e6 - 3.69e5i)T^{2} \) |
| 23 | \( 1 + (-1.50e3 + 1.02e3i)T + (2.35e6 - 5.99e6i)T^{2} \) |
| 29 | \( 1 + (-3.95e3 + 3.67e3i)T + (1.53e6 - 2.04e7i)T^{2} \) |
| 31 | \( 1 + (-7.68e3 - 2.37e3i)T + (2.36e7 + 1.61e7i)T^{2} \) |
| 37 | \( 1 + (-1.58e3 + 2.74e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (783. + 3.43e3i)T + (-1.04e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (-1.44e4 + 6.97e3i)T + (1.42e8 - 1.79e8i)T^{2} \) |
| 53 | \( 1 + (6.31e3 + 1.60e4i)T + (-3.06e8 + 2.84e8i)T^{2} \) |
| 59 | \( 1 + (-3.06e3 + 3.84e3i)T + (-1.59e8 - 6.96e8i)T^{2} \) |
| 61 | \( 1 + (5.36e4 - 1.65e4i)T + (6.97e8 - 4.75e8i)T^{2} \) |
| 67 | \( 1 + (-715. + 9.55e3i)T + (-1.33e9 - 2.01e8i)T^{2} \) |
| 71 | \( 1 + (1.62e4 + 1.10e4i)T + (6.59e8 + 1.67e9i)T^{2} \) |
| 73 | \( 1 + (3.72e3 - 9.48e3i)T + (-1.51e9 - 1.41e9i)T^{2} \) |
| 79 | \( 1 + (-141. - 245. i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (7.64e4 + 7.09e4i)T + (2.94e8 + 3.92e9i)T^{2} \) |
| 89 | \( 1 + (4.98e4 + 4.62e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + (-4.17e4 - 2.01e4i)T + (5.35e9 + 6.71e9i)T^{2} \) |
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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
−15.48929829798821612604145005555, −14.56591714272689503431141833449, −13.51021164767206408049706215279, −11.76318310955452546537594181841, −10.34461478357951579195154923356, −8.727302469508957949557504220199, −7.943643021928700778606039417682, −6.06184993941843713016696237698, −4.88707337561029194212510367199, −2.66648722722443847438512849147,
1.41554203239575676959909759262, 2.78719402939254680015384688460, 4.56235655033298887221502867131, 7.19924856070742634203201688876, 8.027110556037795813891334487133, 10.07237649223313953279379567311, 10.94173455054852647476342624237, 12.29393546183638200998552227307, 13.41400474563349103982539233993, 13.93885246544238809265947918548
