| L(s) = 1 | + (1.11 + 2.31i)2-s + (18.6 − 38.8i)3-s + (35.7 − 44.8i)4-s + (137. + 31.3i)5-s + 110.·6-s + 167. i·7-s + (304. + 69.4i)8-s + (−703. − 881. i)9-s + (80.6 + 353. i)10-s + (−449. − 564. i)11-s + (−1.07e3 − 2.22e3i)12-s + (−269. + 1.18e3i)13-s + (−386. + 186. i)14-s + (3.79e3 − 4.75e3i)15-s + (−639. − 2.80e3i)16-s + (1.34e3 + 5.88e3i)17-s + ⋯ |
| L(s) = 1 | + (0.139 + 0.289i)2-s + (0.692 − 1.43i)3-s + (0.559 − 0.701i)4-s + (1.10 + 0.251i)5-s + 0.512·6-s + 0.487i·7-s + (0.593 + 0.135i)8-s + (−0.964 − 1.20i)9-s + (0.0806 + 0.353i)10-s + (−0.338 − 0.423i)11-s + (−0.621 − 1.28i)12-s + (−0.122 + 0.537i)13-s + (−0.141 + 0.0679i)14-s + (1.12 − 1.40i)15-s + (−0.156 − 0.683i)16-s + (0.273 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.49423 - 1.60689i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.49423 - 1.60689i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.48e4 - 4.59e4i)T \) |
| good | 2 | \( 1 + (-1.11 - 2.31i)T + (-39.9 + 50.0i)T^{2} \) |
| 3 | \( 1 + (-18.6 + 38.8i)T + (-454. - 569. i)T^{2} \) |
| 5 | \( 1 + (-137. - 31.3i)T + (1.40e4 + 6.77e3i)T^{2} \) |
| 7 | \( 1 - 167. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + (449. + 564. i)T + (-3.94e5 + 1.72e6i)T^{2} \) |
| 13 | \( 1 + (269. - 1.18e3i)T + (-4.34e6 - 2.09e6i)T^{2} \) |
| 17 | \( 1 + (-1.34e3 - 5.88e3i)T + (-2.17e7 + 1.04e7i)T^{2} \) |
| 19 | \( 1 + (1.79e3 + 1.42e3i)T + (1.04e7 + 4.58e7i)T^{2} \) |
| 23 | \( 1 + (-1.01e3 - 1.27e3i)T + (-3.29e7 + 1.44e8i)T^{2} \) |
| 29 | \( 1 + (1.21e4 + 2.51e4i)T + (-3.70e8 + 4.65e8i)T^{2} \) |
| 31 | \( 1 + (3.59e4 - 1.73e4i)T + (5.53e8 - 6.93e8i)T^{2} \) |
| 37 | \( 1 - 5.76e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-3.84e4 + 1.85e4i)T + (2.96e9 - 3.71e9i)T^{2} \) |
| 47 | \( 1 + (7.12e4 - 8.93e4i)T + (-2.39e9 - 1.05e10i)T^{2} \) |
| 53 | \( 1 + (-4.38e4 - 1.92e5i)T + (-1.99e10 + 9.61e9i)T^{2} \) |
| 59 | \( 1 + (8.72e4 + 3.82e5i)T + (-3.80e10 + 1.83e10i)T^{2} \) |
| 61 | \( 1 + (1.31e5 - 2.72e5i)T + (-3.21e10 - 4.02e10i)T^{2} \) |
| 67 | \( 1 + (-2.24e5 + 2.81e5i)T + (-2.01e10 - 8.81e10i)T^{2} \) |
| 71 | \( 1 + (9.10e4 + 7.26e4i)T + (2.85e10 + 1.24e11i)T^{2} \) |
| 73 | \( 1 + (-3.50e5 - 7.99e4i)T + (1.36e11 + 6.56e10i)T^{2} \) |
| 79 | \( 1 - 6.48e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + (-2.85e5 - 1.37e5i)T + (2.03e11 + 2.55e11i)T^{2} \) |
| 89 | \( 1 + (5.31e5 - 1.10e6i)T + (-3.09e11 - 3.88e11i)T^{2} \) |
| 97 | \( 1 + (1.05e6 + 1.31e6i)T + (-1.85e11 + 8.12e11i)T^{2} \) |
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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
−14.26172036885305065209774172070, −13.56577900888208008234776115496, −12.45428210910122438874412772177, −10.89470061585265620610086583635, −9.400465303222808412513685130326, −7.907232727742976024994332862905, −6.56207535914851808379479234916, −5.78717986402114799913116860611, −2.45815631895908165785791057324, −1.52507041090639818536956776064,
2.36290556556657323774216441636, 3.74601578192844233794595649325, 5.24611751809204397462987848802, 7.45339936813215043568493613181, 9.048030940113261126690333671762, 10.04821792886679249356279327164, 10.96141724999999310492152622903, 12.70869988690273694117613527697, 13.79760297018300147442395005522, 14.92086353181596059246494498524
