L(s) = 1 | + (1.80e4 + 6.29e4i)2-s + (−3.64e9 + 2.27e9i)4-s − 1.21e11·5-s + 3.17e10i·7-s + (−2.09e14 − 1.88e14i)8-s + (−2.18e15 − 7.63e15i)10-s + 6.68e16i·11-s + 5.92e17·13-s + (−1.99e15 + 5.72e14i)14-s + (8.09e18 − 1.65e19i)16-s − 4.82e19·17-s + 9.57e19i·19-s + (4.41e20 − 2.75e20i)20-s + (−4.21e21 + 1.20e21i)22-s − 5.33e20i·23-s + ⋯ |
L(s) = 1 | + (0.275 + 0.961i)2-s + (−0.848 + 0.529i)4-s − 0.794·5-s + 0.000954i·7-s + (−0.742 − 0.669i)8-s + (−0.218 − 0.763i)10-s + 1.45i·11-s + 0.889·13-s + (−0.000917 + 0.000263i)14-s + (0.438 − 0.898i)16-s − 0.992·17-s + 0.331i·19-s + (0.673 − 0.421i)20-s + (−1.39 + 0.400i)22-s − 0.0869i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(33-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{33}{2})\) |
\(\approx\) |
\(1.377893577\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.377893577\) |
\(L(17)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.80e4 - 6.29e4i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.21e11T + 2.32e22T^{2} \) |
| 7 | \( 1 - 3.17e10iT - 1.10e27T^{2} \) |
| 11 | \( 1 - 6.68e16iT - 2.11e33T^{2} \) |
| 13 | \( 1 - 5.92e17T + 4.42e35T^{2} \) |
| 17 | \( 1 + 4.82e19T + 2.36e39T^{2} \) |
| 19 | \( 1 - 9.57e19iT - 8.31e40T^{2} \) |
| 23 | \( 1 + 5.33e20iT - 3.76e43T^{2} \) |
| 29 | \( 1 - 4.06e23T + 6.26e46T^{2} \) |
| 31 | \( 1 + 4.12e23iT - 5.29e47T^{2} \) |
| 37 | \( 1 - 1.13e25T + 1.52e50T^{2} \) |
| 41 | \( 1 - 2.76e23T + 4.06e51T^{2} \) |
| 43 | \( 1 - 1.68e26iT - 1.86e52T^{2} \) |
| 47 | \( 1 - 7.88e26iT - 3.21e53T^{2} \) |
| 53 | \( 1 - 5.53e27T + 1.50e55T^{2} \) |
| 59 | \( 1 - 4.25e28iT - 4.64e56T^{2} \) |
| 61 | \( 1 - 5.00e28T + 1.35e57T^{2} \) |
| 67 | \( 1 + 1.33e29iT - 2.71e58T^{2} \) |
| 71 | \( 1 - 5.68e29iT - 1.73e59T^{2} \) |
| 73 | \( 1 + 6.17e29T + 4.22e59T^{2} \) |
| 79 | \( 1 - 1.44e30iT - 5.29e60T^{2} \) |
| 83 | \( 1 + 6.98e30iT - 2.57e61T^{2} \) |
| 89 | \( 1 - 1.07e31T + 2.40e62T^{2} \) |
| 97 | \( 1 + 3.35e31T + 3.77e63T^{2} \) |
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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
−11.56281458739190251580922179164, −10.03917624166267096286648253619, −8.826199468604781853886782548806, −7.86739408375511821895018189819, −6.97442321756749859296906431645, −5.96224793415634875544183260786, −4.50152618504532957527658103619, −4.07986103876724545174510778359, −2.61416575170829676364370102877, −0.990275197684671596046007328124,
0.30931442121464625536319499503, 0.935172790955911033595409964134, 2.34723229376907078804220974631, 3.43002468989799757969790703185, 4.12682590156424892370721844162, 5.38985543489131254496605828547, 6.54643527428625248458117495723, 8.299312084709050695779487939690, 8.860086952949654998352301293828, 10.41528404283800496299766540933