| L(s) = 1 | + (1.16 − 1.62i)2-s + (−2.99 + 0.218i)3-s + (−1.26 − 3.79i)4-s + (−2.90 − 5.03i)5-s + (−3.14 + 5.10i)6-s + (−0.363 + 0.629i)7-s + (−7.63 − 2.38i)8-s + (8.90 − 1.30i)9-s + (−11.5 − 1.17i)10-s + (2.03 − 3.51i)11-s + (4.61 + 11.0i)12-s + (13.3 − 7.70i)13-s + (0.596 + 1.32i)14-s + (9.79 + 14.4i)15-s + (−12.8 + 9.59i)16-s − 11.6i·17-s + ⋯ |
| L(s) = 1 | + (0.584 − 0.811i)2-s + (−0.997 + 0.0727i)3-s + (−0.316 − 0.948i)4-s + (−0.581 − 1.00i)5-s + (−0.524 + 0.851i)6-s + (−0.0519 + 0.0899i)7-s + (−0.954 − 0.298i)8-s + (0.989 − 0.145i)9-s + (−1.15 − 0.117i)10-s + (0.184 − 0.319i)11-s + (0.384 + 0.923i)12-s + (1.02 − 0.592i)13-s + (0.0425 + 0.0946i)14-s + (0.653 + 0.962i)15-s + (−0.800 + 0.599i)16-s − 0.682i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.754 + 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.358117 - 0.957924i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.358117 - 0.957924i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.16 + 1.62i)T \) |
| 3 | \( 1 + (2.99 - 0.218i)T \) |
| good | 5 | \( 1 + (2.90 + 5.03i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (0.363 - 0.629i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-2.03 + 3.51i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-13.3 + 7.70i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 11.6iT - 289T^{2} \) |
| 19 | \( 1 - 35.6iT - 361T^{2} \) |
| 23 | \( 1 + (-26.0 + 15.0i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-16.7 + 28.9i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (17.8 + 30.8i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 24.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (49.7 - 28.7i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (6.79 + 3.92i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (4.39 + 2.53i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 28.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + (19.8 + 34.3i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-33.9 - 19.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-63.9 + 36.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 88.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 105.T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-35.5 + 61.5i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-18.1 + 31.4i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 121. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (25.3 - 43.8i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.54627326308798877751743715405, −12.60837204226322065828350223038, −11.88036037708484328935145494797, −10.94779343777054962717955390305, −9.780680118532489013344055527346, −8.329079991405021125440726725044, −6.21746174661305233188977623599, −5.09556992625894510650811520771, −3.85244765469718415006586654959, −0.922192066749091982319525872741,
3.62881865462350245501770274450, 5.10738248150439273220172962598, 6.71551006239772359018625273246, 7.08908724594421915245152958506, 8.886413666803435481461638578899, 10.80489496986105986050763489965, 11.50082396952709641180174909414, 12.72415548335932264427899827628, 13.75798741180356454635053016728, 15.08949075109080550988921833343
