| L(s) = 1 | + (−6.47 − 4.70i)2-s + (66.4 − 48.2i)3-s + (19.7 + 60.8i)4-s − 95.3·5-s − 656.·6-s + (418. + 1.28e3i)7-s + (158. − 486. i)8-s + (1.40e3 − 4.33e3i)9-s + (617. + 448. i)10-s + (−2.56e3 − 7.88e3i)11-s + (4.25e3 + 3.08e3i)12-s + (5.74e3 − 4.17e3i)13-s + (3.34e3 − 1.03e4i)14-s + (−6.33e3 + 4.60e3i)15-s + (−3.31e3 + 2.40e3i)16-s + (3.05e3 − 9.41e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (1.42 − 1.03i)3-s + (0.154 + 0.475i)4-s − 0.341·5-s − 1.24·6-s + (0.461 + 1.41i)7-s + (0.109 − 0.336i)8-s + (0.643 − 1.98i)9-s + (0.195 + 0.141i)10-s + (−0.580 − 1.78i)11-s + (0.710 + 0.516i)12-s + (0.724 − 0.526i)13-s + (0.326 − 1.00i)14-s + (−0.484 + 0.352i)15-s + (−0.202 + 0.146i)16-s + (0.150 − 0.464i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.12078 - 1.83769i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12078 - 1.83769i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (6.47 + 4.70i)T \) |
| 31 | \( 1 + (1.64e5 + 2.42e4i)T \) |
| good | 3 | \( 1 + (-66.4 + 48.2i)T + (675. - 2.07e3i)T^{2} \) |
| 5 | \( 1 + 95.3T + 7.81e4T^{2} \) |
| 7 | \( 1 + (-418. - 1.28e3i)T + (-6.66e5 + 4.84e5i)T^{2} \) |
| 11 | \( 1 + (2.56e3 + 7.88e3i)T + (-1.57e7 + 1.14e7i)T^{2} \) |
| 13 | \( 1 + (-5.74e3 + 4.17e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-3.05e3 + 9.41e3i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (-1.10e4 - 8.06e3i)T + (2.76e8 + 8.50e8i)T^{2} \) |
| 23 | \( 1 + (-3.02e4 + 9.31e4i)T + (-2.75e9 - 2.00e9i)T^{2} \) |
| 29 | \( 1 + (-3.67e4 - 2.67e4i)T + (5.33e9 + 1.64e10i)T^{2} \) |
| 37 | \( 1 - 4.50e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (3.92e5 + 2.85e5i)T + (6.01e10 + 1.85e11i)T^{2} \) |
| 43 | \( 1 + (-4.69e5 - 3.41e5i)T + (8.39e10 + 2.58e11i)T^{2} \) |
| 47 | \( 1 + (-1.81e5 + 1.31e5i)T + (1.56e11 - 4.81e11i)T^{2} \) |
| 53 | \( 1 + (2.53e5 - 7.81e5i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (1.27e6 - 9.29e5i)T + (7.69e11 - 2.36e12i)T^{2} \) |
| 61 | \( 1 + 1.09e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 2.96e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (1.02e6 - 3.15e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 + (6.25e5 + 1.92e6i)T + (-8.93e12 + 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-1.12e6 + 3.47e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (-6.92e6 - 5.03e6i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + (2.79e5 + 8.59e5i)T + (-3.57e13 + 2.59e13i)T^{2} \) |
| 97 | \( 1 + (-1.27e6 - 3.92e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−13.11967430256901307628335801801, −12.15150733248485472509031914675, −11.01036906444711436542404455423, −9.094012357795185558995939298205, −8.441976315546223059054226944024, −7.78283407100092014766186168843, −5.97806453674392983586737706397, −3.26005154136906744698154279074, −2.41244656113878492840098613095, −0.837340516747519863244921744945,
1.75232981667264500615796628475, 3.73233878758080283040042863973, 4.71819329635608271051911963767, 7.36256584919333796483486625038, 7.892621694156718814617303456432, 9.343395272339786647021405029480, 10.06311861871816362472430686190, 11.10744259854652205927620011319, 13.28525594065024911223517089643, 14.18369229129810983062557084541
