| L(s) = 1 | + (4.5 + 7.79i)3-s + (3 − 5.19i)5-s + (−59.5 − 115. i)7-s + (−40.5 + 70.1i)9-s + (−333 − 576. i)11-s − 559·13-s + 54·15-s + (870 + 1.50e3i)17-s + (578.5 − 1.00e3i)19-s + (630 − 982. i)21-s + (−1.73e3 + 3.00e3i)23-s + (1.54e3 + 2.67e3i)25-s − 729·27-s + 3.37e3·29-s + (3.14e3 + 5.44e3i)31-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.499i)3-s + (0.0536 − 0.0929i)5-s + (−0.458 − 0.888i)7-s + (−0.166 + 0.288i)9-s + (−0.829 − 1.43i)11-s − 0.917·13-s + 0.0619·15-s + (0.730 + 1.26i)17-s + (0.367 − 0.636i)19-s + (0.311 − 0.485i)21-s + (−0.683 + 1.18i)23-s + (0.494 + 0.856i)25-s − 0.192·27-s + 0.744·29-s + (0.588 + 1.01i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0165 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.301468426\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.301468426\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 + (59.5 + 115. i)T \) |
| good | 5 | \( 1 + (-3 + 5.19i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (333 + 576. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 559T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-870 - 1.50e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-578.5 + 1.00e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (1.73e3 - 3.00e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 - 3.37e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-3.14e3 - 5.44e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (1.56e3 - 2.71e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 4.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.14e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.15e3 + 2.00e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-1.41e4 - 2.45e4i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-1.02e4 - 1.77e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-2.31e3 + 4.00e3i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (9.37e3 + 1.62e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 3.82e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (3.52e4 + 6.11e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.11e4 - 5.39e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 7.98e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-9.06e3 + 1.56e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 1.24e5T + 8.58e9T^{2} \) |
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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
−10.62489898153418921015971785496, −10.22595783769476737468305953660, −9.143520093984145713606732767018, −8.140414292757208274461740084292, −7.30418845976713413572920226417, −5.97216077023172006266100781178, −5.00086807075000355526953361316, −3.68510052302968977127038674825, −2.89868960803416599973635966574, −1.05022587609681816431050567968,
0.36042115180128449192186464010, 2.22456572035789607936634683002, 2.77771648441418477519222978630, 4.55634796869498295449274273419, 5.57479463435077532091349417716, 6.77507210299774285691148234110, 7.59119896798043885954714259424, 8.514231459579711460744743713197, 9.825410217497097562667727088022, 10.04493941833510751799712881080
