| L(s) = 1 | + (−0.257 − 1.12i)2-s + (−18.5 − 17.1i)3-s + (27.6 − 13.3i)4-s + (37.0 − 5.57i)5-s + (−14.5 + 25.2i)6-s + (−53.8 − 93.3i)7-s + (−45.1 − 56.6i)8-s + (29.4 + 392. i)9-s + (−15.7 − 40.2i)10-s + (−544. − 262. i)11-s + (−739. − 228. i)12-s + (−393. + 1.00e3i)13-s + (−91.2 + 84.6i)14-s + (−780. − 532. i)15-s + (559. − 701. i)16-s + (905. + 136. i)17-s + ⋯ |
| L(s) = 1 | + (−0.0454 − 0.199i)2-s + (−1.18 − 1.10i)3-s + (0.863 − 0.415i)4-s + (0.662 − 0.0997i)5-s + (−0.165 + 0.286i)6-s + (−0.415 − 0.719i)7-s + (−0.249 − 0.312i)8-s + (0.121 + 1.61i)9-s + (−0.0499 − 0.127i)10-s + (−1.35 − 0.653i)11-s + (−1.48 − 0.457i)12-s + (−0.645 + 1.64i)13-s + (−0.124 + 0.115i)14-s + (−0.895 − 0.610i)15-s + (0.546 − 0.685i)16-s + (0.760 + 0.114i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0733i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0347442 - 0.945504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0347442 - 0.945504i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.31e3 - 1.19e4i)T \) |
| good | 2 | \( 1 + (0.257 + 1.12i)T + (-28.8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (18.5 + 17.1i)T + (18.1 + 242. i)T^{2} \) |
| 5 | \( 1 + (-37.0 + 5.57i)T + (2.98e3 - 921. i)T^{2} \) |
| 7 | \( 1 + (53.8 + 93.3i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (544. + 262. i)T + (1.00e5 + 1.25e5i)T^{2} \) |
| 13 | \( 1 + (393. - 1.00e3i)T + (-2.72e5 - 2.52e5i)T^{2} \) |
| 17 | \( 1 + (-905. - 136. i)T + (1.35e6 + 4.18e5i)T^{2} \) |
| 19 | \( 1 + (-139. + 1.86e3i)T + (-2.44e6 - 3.69e5i)T^{2} \) |
| 23 | \( 1 + (275. - 187. i)T + (2.35e6 - 5.99e6i)T^{2} \) |
| 29 | \( 1 + (-3.40e3 + 3.15e3i)T + (1.53e6 - 2.04e7i)T^{2} \) |
| 31 | \( 1 + (-130. - 40.2i)T + (2.36e7 + 1.61e7i)T^{2} \) |
| 37 | \( 1 + (-3.67e3 + 6.35e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + (3.98e3 + 1.74e4i)T + (-1.04e8 + 5.02e7i)T^{2} \) |
| 47 | \( 1 + (-1.74e3 + 841. i)T + (1.42e8 - 1.79e8i)T^{2} \) |
| 53 | \( 1 + (4.33e3 + 1.10e4i)T + (-3.06e8 + 2.84e8i)T^{2} \) |
| 59 | \( 1 + (-2.60e4 + 3.26e4i)T + (-1.59e8 - 6.96e8i)T^{2} \) |
| 61 | \( 1 + (-4.55e3 + 1.40e3i)T + (6.97e8 - 4.75e8i)T^{2} \) |
| 67 | \( 1 + (2.17e3 - 2.90e4i)T + (-1.33e9 - 2.01e8i)T^{2} \) |
| 71 | \( 1 + (1.88e4 + 1.28e4i)T + (6.59e8 + 1.67e9i)T^{2} \) |
| 73 | \( 1 + (-8.08e3 + 2.05e4i)T + (-1.51e9 - 1.41e9i)T^{2} \) |
| 79 | \( 1 + (-4.67e4 - 8.10e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (7.83e4 + 7.26e4i)T + (2.94e8 + 3.92e9i)T^{2} \) |
| 89 | \( 1 + (-6.67e4 - 6.18e4i)T + (4.17e8 + 5.56e9i)T^{2} \) |
| 97 | \( 1 + (-1.51e4 - 7.28e3i)T + (5.35e9 + 6.71e9i)T^{2} \) |
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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
−13.99941713269550131996801199403, −13.06752256081734354439666260087, −11.87324890390959513836524878157, −10.97529121869311879363762226075, −9.869696020830814757460919847614, −7.41624813024768274093565278588, −6.52058421782903173392941342673, −5.40957737122860967030914072322, −2.17259290704335904769779279670, −0.55990190690516080847956208575,
2.87149753480109782158628834158, 5.28706169547313294091148410249, 6.00259725964059652773232855818, 7.84410084293132537450382955807, 10.05833773884754626134933770369, 10.37134325312275461481339052725, 11.96018901956285206293443133285, 12.72155787547365732760346446759, 14.98934879128893963151571159054, 15.66112764761943606849423677623
