| L(s) = 1 | + (−7.04 + 8.82i)2-s + (3.19 + 4.01i)3-s + (−21.2 − 93.1i)4-s + (78.2 + 37.7i)5-s − 57.9·6-s + 141.·7-s + (646. + 311. i)8-s + (48.2 − 211. i)9-s + (−884. + 425. i)10-s + (53.0 − 232. i)11-s + (305. − 383. i)12-s + (−64.9 − 31.2i)13-s + (−998. + 1.25e3i)14-s + (99.2 + 434. i)15-s + (−4.54e3 + 2.18e3i)16-s + (962. − 463. i)17-s + ⋯ |
| L(s) = 1 | + (−1.24 + 1.56i)2-s + (0.205 + 0.257i)3-s + (−0.664 − 2.91i)4-s + (1.40 + 0.674i)5-s − 0.656·6-s + 1.09·7-s + (3.56 + 1.71i)8-s + (0.198 − 0.869i)9-s + (−2.79 + 1.34i)10-s + (0.132 − 0.579i)11-s + (0.612 − 0.767i)12-s + (−0.106 − 0.0513i)13-s + (−1.36 + 1.70i)14-s + (0.113 + 0.498i)15-s + (−4.43 + 2.13i)16-s + (0.807 − 0.388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.819951 + 0.970221i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.819951 + 0.970221i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-193. - 1.21e4i)T \) |
| good | 2 | \( 1 + (7.04 - 8.82i)T + (-7.12 - 31.1i)T^{2} \) |
| 3 | \( 1 + (-3.19 - 4.01i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (-78.2 - 37.7i)T + (1.94e3 + 2.44e3i)T^{2} \) |
| 7 | \( 1 - 141.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-53.0 + 232. i)T + (-1.45e5 - 6.98e4i)T^{2} \) |
| 13 | \( 1 + (64.9 + 31.2i)T + (2.31e5 + 2.90e5i)T^{2} \) |
| 17 | \( 1 + (-962. + 463. i)T + (8.85e5 - 1.11e6i)T^{2} \) |
| 19 | \( 1 + (-56.0 - 245. i)T + (-2.23e6 + 1.07e6i)T^{2} \) |
| 23 | \( 1 + (485. - 2.12e3i)T + (-5.79e6 - 2.79e6i)T^{2} \) |
| 29 | \( 1 + (2.93e3 - 3.67e3i)T + (-4.56e6 - 1.99e7i)T^{2} \) |
| 31 | \( 1 + (-1.29e3 + 1.62e3i)T + (-6.37e6 - 2.79e7i)T^{2} \) |
| 37 | \( 1 + 3.87e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-4.83e3 + 6.06e3i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 47 | \( 1 + (-15.4 - 67.7i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (3.40e3 - 1.63e3i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 + (2.10e4 - 1.01e4i)T + (4.45e8 - 5.58e8i)T^{2} \) |
| 61 | \( 1 + (2.58e4 + 3.23e4i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (3.04e3 + 1.33e4i)T + (-1.21e9 + 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-5.40e3 - 2.36e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (4.34e3 + 2.09e3i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 - 3.09e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-8.96e3 - 1.12e4i)T + (-8.76e8 + 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-5.13e4 - 6.43e4i)T + (-1.24e9 + 5.44e9i)T^{2} \) |
| 97 | \( 1 + (2.78e4 - 1.22e5i)T + (-7.73e9 - 3.72e9i)T^{2} \) |
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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
−15.25268400168685293170388342149, −14.41376636232485739274198210155, −13.88027330113524453027798351950, −10.96857312067761977451039928822, −9.861772419956178318903339886496, −9.089389550855974320563444518693, −7.68659348668072844648401943292, −6.35860699650025682858895388480, −5.33356700864401444998317672148, −1.41926793246155071557714966362,
1.40887770727761407309369574404, 2.21083709767826275861614442545, 4.70985324934160778542561528285, 7.68031783348137275134034291484, 8.704956911683085372949553819051, 9.845310016452844013150011165394, 10.71151392973576350291142586840, 12.11259663552450692166802092021, 13.08093764594225175109653755928, 14.01861241391535701641210590126
