| L(s) = 1 | + (0.185 − 0.385i)2-s + (−2.30 + 3.37i)3-s + (2.38 + 2.98i)4-s + (−0.146 − 0.157i)5-s + (0.873 + 1.51i)6-s + (0.871 + 0.503i)7-s + (3.25 − 0.743i)8-s + (−2.81 − 7.16i)9-s + (−0.0877 + 0.0270i)10-s + (10.2 − 12.8i)11-s + (−15.5 + 1.16i)12-s + (−8.32 − 2.56i)13-s + (0.355 − 0.242i)14-s + (0.868 − 0.130i)15-s + (−3.07 + 13.4i)16-s + (−11.3 − 10.5i)17-s + ⋯ |
| L(s) = 1 | + (0.0927 − 0.192i)2-s + (−0.767 + 1.12i)3-s + (0.595 + 0.746i)4-s + (−0.0292 − 0.0315i)5-s + (0.145 + 0.252i)6-s + (0.124 + 0.0719i)7-s + (0.407 − 0.0929i)8-s + (−0.312 − 0.796i)9-s + (−0.00877 + 0.00270i)10-s + (0.935 − 1.17i)11-s + (−1.29 + 0.0971i)12-s + (−0.640 − 0.197i)13-s + (0.0253 − 0.0173i)14-s + (0.0578 − 0.00872i)15-s + (−0.192 + 0.843i)16-s + (−0.667 − 0.619i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.895720 + 0.534589i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.895720 + 0.534589i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (26.1 - 34.1i)T \) |
| good | 2 | \( 1 + (-0.185 + 0.385i)T + (-2.49 - 3.12i)T^{2} \) |
| 3 | \( 1 + (2.30 - 3.37i)T + (-3.28 - 8.37i)T^{2} \) |
| 5 | \( 1 + (0.146 + 0.157i)T + (-1.86 + 24.9i)T^{2} \) |
| 7 | \( 1 + (-0.871 - 0.503i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-10.2 + 12.8i)T + (-26.9 - 117. i)T^{2} \) |
| 13 | \( 1 + (8.32 + 2.56i)T + (139. + 95.2i)T^{2} \) |
| 17 | \( 1 + (11.3 + 10.5i)T + (21.5 + 288. i)T^{2} \) |
| 19 | \( 1 + (-18.4 - 7.22i)T + (264. + 245. i)T^{2} \) |
| 23 | \( 1 + (-31.0 - 4.68i)T + (505. + 155. i)T^{2} \) |
| 29 | \( 1 + (3.63 + 5.33i)T + (-307. + 782. i)T^{2} \) |
| 31 | \( 1 + (-1.13 - 15.1i)T + (-950. + 143. i)T^{2} \) |
| 37 | \( 1 + (47.5 - 27.4i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (38.6 + 18.6i)T + (1.04e3 + 1.31e3i)T^{2} \) |
| 47 | \( 1 + (46.0 + 57.7i)T + (-491. + 2.15e3i)T^{2} \) |
| 53 | \( 1 + (-34.2 + 10.5i)T + (2.32e3 - 1.58e3i)T^{2} \) |
| 59 | \( 1 + (10.8 - 47.3i)T + (-3.13e3 - 1.51e3i)T^{2} \) |
| 61 | \( 1 + (-104. - 7.86i)T + (3.67e3 + 554. i)T^{2} \) |
| 67 | \( 1 + (13.7 - 35.1i)T + (-3.29e3 - 3.05e3i)T^{2} \) |
| 71 | \( 1 + (9.75 + 64.6i)T + (-4.81e3 + 1.48e3i)T^{2} \) |
| 73 | \( 1 + (12.1 - 39.2i)T + (-4.40e3 - 3.00e3i)T^{2} \) |
| 79 | \( 1 + (-45.0 + 78.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-26.6 - 18.1i)T + (2.51e3 + 6.41e3i)T^{2} \) |
| 89 | \( 1 + (68.3 - 100. i)T + (-2.89e3 - 7.37e3i)T^{2} \) |
| 97 | \( 1 + (-30.4 + 38.2i)T + (-2.09e3 - 9.17e3i)T^{2} \) |
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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
−16.23468293599572385508439606680, −15.10200544285744405547451867166, −13.56305297710964400801538792260, −11.88074191879752189493990931327, −11.36381796554126233185780307730, −10.14052065196831970115536397516, −8.628873989089430371670824809344, −6.79927193436461636783892977603, −5.07113223948285871624877828040, −3.44997307779222143874972327012,
1.61767886203654971340287842333, 5.13338182222440853465182388504, 6.69572259322019674894179997326, 7.22993398568239611841506309259, 9.513607104965123056460934391290, 11.07464379488799481052262382171, 11.94454413943990613140245897776, 13.05995851096864760040680716408, 14.48605308275749422341637995804, 15.39762377461142292826092839827
