| L(s) = 1 | + (2.25 + 11.0i)2-s + (29.8 − 71.9i)3-s + (−117. + 50.0i)4-s + (−87.5 + 36.2i)5-s + (865. + 168. i)6-s + (226. − 226. i)7-s + (−820. − 1.19e3i)8-s + (−2.74e3 − 2.74e3i)9-s + (−599. − 889. i)10-s + (−2.09e3 − 5.05e3i)11-s + (86.8 + 9.97e3i)12-s + (−6.64e3 − 2.75e3i)13-s + (3.02e3 + 2.00e3i)14-s + 7.38e3i·15-s + (1.13e4 − 1.17e4i)16-s − 2.78e4i·17-s + ⋯ |
| L(s) = 1 | + (0.199 + 0.979i)2-s + (0.637 − 1.53i)3-s + (−0.920 + 0.390i)4-s + (−0.313 + 0.129i)5-s + (1.63 + 0.317i)6-s + (0.249 − 0.249i)7-s + (−0.566 − 0.824i)8-s + (−1.25 − 1.25i)9-s + (−0.189 − 0.281i)10-s + (−0.473 − 1.14i)11-s + (0.0145 + 1.66i)12-s + (−0.839 − 0.347i)13-s + (0.294 + 0.194i)14-s + 0.564i·15-s + (0.694 − 0.719i)16-s − 1.37i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0217 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.0217 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.06581 - 1.04286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06581 - 1.04286i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.25 - 11.0i)T \) |
| good | 3 | \( 1 + (-29.8 + 71.9i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (87.5 - 36.2i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-226. + 226. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (2.09e3 + 5.05e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (6.64e3 + 2.75e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 2.78e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-2.15e4 - 8.93e3i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-4.55e4 - 4.55e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (1.75e4 - 4.24e4i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.22e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-5.65e5 + 2.34e5i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (2.92e3 + 2.92e3i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (2.79e5 + 6.73e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 3.42e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-5.43e5 - 1.31e6i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-1.93e6 + 8.00e5i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (-1.04e6 + 2.52e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (1.50e6 - 3.63e6i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (8.28e4 - 8.28e4i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (2.21e6 + 2.21e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 + 7.09e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-9.42e5 - 3.90e5i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-6.11e6 + 6.11e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 + 2.30e6T + 8.07e13T^{2} \) |
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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
−14.67565570514942799237822288755, −13.77343028247345933765053165165, −13.00273153419391012478499336425, −11.64717476111395298059183306469, −9.162400683436771333448495154496, −7.73489752630147346251113726338, −7.28727695364786766519337238078, −5.55914005414265231623716688251, −3.08601133803899492104938338303, −0.61576093031909475789358056735,
2.45121470036118687838008477782, 4.07632159177523565749734230660, 5.01191548378223203382893146650, 8.262390608579917120258581537845, 9.552274796725493629090924014635, 10.29878722033567691275444904824, 11.63090697371687713933003260614, 13.02445403890129701722839517719, 14.76453295514449844449285232283, 15.00456213808145528895755859184
