| L(s) = 1 | + (7.98 + 34.9i)2-s + (139. + 129. i)3-s + (−699. + 336. i)4-s + (2.45e3 − 370. i)5-s + (−3.42e3 + 5.93e3i)6-s + (−493. − 854. i)7-s + (−5.91e3 − 7.42e3i)8-s + (1.24e3 + 1.66e4i)9-s + (3.25e4 + 8.30e4i)10-s + (−2.79e4 − 1.34e4i)11-s + (−1.41e5 − 4.36e4i)12-s + (−4.13e4 + 1.05e5i)13-s + (2.59e4 − 2.40e4i)14-s + (3.91e5 + 2.67e5i)15-s + (−3.53e4 + 4.43e4i)16-s + (−3.01e5 − 4.54e4i)17-s + ⋯ |
| L(s) = 1 | + (0.353 + 1.54i)2-s + (0.996 + 0.925i)3-s + (−1.36 + 0.658i)4-s + (1.75 − 0.265i)5-s + (−1.07 + 1.86i)6-s + (−0.0776 − 0.134i)7-s + (−0.510 − 0.640i)8-s + (0.0634 + 0.847i)9-s + (1.03 + 2.62i)10-s + (−0.575 − 0.276i)11-s + (−1.97 − 0.607i)12-s + (−0.401 + 1.02i)13-s + (0.180 − 0.167i)14-s + (1.99 + 1.36i)15-s + (−0.135 + 0.169i)16-s + (−0.875 − 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0388i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0388i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.0752973 + 3.87113i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0752973 + 3.87113i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.78e6 - 2.23e7i)T \) |
| good | 2 | \( 1 + (-7.98 - 34.9i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (-139. - 129. i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (-2.45e3 + 370. i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (493. + 854. i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (2.79e4 + 1.34e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (4.13e4 - 1.05e5i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (3.01e5 + 4.54e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (5.69e4 - 7.59e5i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (-1.11e6 + 7.63e5i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (-1.25e6 + 1.16e6i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (-1.44e6 - 4.44e5i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (-2.55e6 + 4.43e6i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (4.40e6 + 1.93e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (-3.28e7 + 1.58e7i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (-2.08e7 - 5.30e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (-2.24e7 + 2.81e7i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (-1.28e8 + 3.95e7i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (-1.85e7 + 2.47e8i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (-2.57e8 - 1.75e8i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (1.58e8 - 4.03e8i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (1.65e8 + 2.87e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (3.55e8 + 3.29e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (7.25e8 + 6.73e8i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (-1.01e9 - 4.88e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
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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
−14.39703490513256694482322949112, −14.02237933860789704606846126750, −13.03410825362219038314442986863, −10.33910864293974118262923273948, −9.280985941656765162311885660960, −8.486820640812508188163951442756, −6.73427469895986383354334309360, −5.52252024279699153011227132306, −4.33550130251909464912196377227, −2.30372682776429361690250863926,
1.13566008422492692128980550780, 2.42192433181962728634848098826, 2.74261106175001753937581065470, 5.16558973182945854137935946910, 6.97939259728016732646410622328, 8.868993463938302160357270159906, 9.889176435354836517900829074241, 10.91149600532394265592220226555, 12.70595555194766881724837165897, 13.26093416589408383311577917547
