| L(s) = 1 | + (2.44 + 1.41i)2-s + (−8.98 − 0.578i)3-s + (3.99 + 6.92i)4-s + (−35.7 − 20.6i)5-s + (−21.1 − 14.1i)6-s + (−9.50 − 48.0i)7-s + 22.6i·8-s + (80.3 + 10.3i)9-s + (−58.3 − 101. i)10-s + (−127. + 73.4i)11-s + (−31.9 − 64.5i)12-s − 89.0·13-s + (44.6 − 131. i)14-s + (309. + 205. i)15-s + (−32.0 + 55.4i)16-s + (41.5 − 23.9i)17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.997 − 0.0642i)3-s + (0.249 + 0.433i)4-s + (−1.42 − 0.825i)5-s + (−0.588 − 0.392i)6-s + (−0.194 − 0.980i)7-s + 0.353i·8-s + (0.991 + 0.128i)9-s + (−0.583 − 1.01i)10-s + (−1.05 + 0.607i)11-s + (−0.221 − 0.448i)12-s − 0.527·13-s + (0.227 − 0.669i)14-s + (1.37 + 0.915i)15-s + (−0.125 + 0.216i)16-s + (0.143 − 0.0829i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.134696 - 0.358945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.134696 - 0.358945i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.44 - 1.41i)T \) |
| 3 | \( 1 + (8.98 + 0.578i)T \) |
| 7 | \( 1 + (9.50 + 48.0i)T \) |
| good | 5 | \( 1 + (35.7 + 20.6i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (127. - 73.4i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + 89.0T + 2.85e4T^{2} \) |
| 17 | \( 1 + (-41.5 + 23.9i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-30.9 + 53.6i)T + (-6.51e4 - 1.12e5i)T^{2} \) |
| 23 | \( 1 + (127. + 73.7i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + 1.07e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + (-524. - 909. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-1.05e3 + 1.82e3i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + 1.69e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 3.63e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (1.91e3 + 1.10e3i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-2.85e3 + 1.64e3i)T + (3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (1.28e3 - 744. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (1.51e3 - 2.62e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-2.27e3 - 3.93e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 155. iT - 2.54e7T^{2} \) |
| 73 | \( 1 + (2.31e3 + 4.01e3i)T + (-1.41e7 + 2.45e7i)T^{2} \) |
| 79 | \( 1 + (2.35e3 - 4.08e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 - 2.60e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (-6.52e3 - 3.76e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 5.57e3T + 8.85e7T^{2} \) |
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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
−15.17046875714216358506377475121, −13.33149646760678902029053800994, −12.44499907506497464865673246486, −11.56569092814869169372943734750, −10.22410226770240402836630665736, −7.934823096849376233241390142469, −7.06373540778841925437000706282, −5.08969355538935523024500491941, −4.11748355253259378861293776628, −0.22519168774728030055754493711,
3.14555578796449890587312804935, 4.92656259830060628196341960938, 6.38867404763478826041211806921, 7.88594376702172011018640136610, 10.14808650239934007118594672649, 11.31601735218157586216499278317, 11.90959365420026936399306139096, 13.01767368330622821887289876677, 14.88497847066465532808385376047, 15.55776549519136354350421566036
