| L(s) = 1 | + (−1.31 − 1.18i)3-s + (2.72 − 4.71i)5-s + (−0.568 + 5.40i)7-s + (−0.615 − 5.85i)9-s + (−8.24 − 18.5i)11-s + (0.300 − 1.41i)13-s + (−9.14 + 2.97i)15-s + (8.32 − 18.6i)17-s + (9.86 − 2.09i)19-s + (7.13 − 6.42i)21-s + (21.5 + 29.6i)23-s + (−2.33 − 4.04i)25-s + (−15.4 + 21.2i)27-s + (−33.4 − 10.8i)29-s + (−16.7 + 26.1i)31-s + ⋯ |
| L(s) = 1 | + (−0.437 − 0.393i)3-s + (0.544 − 0.943i)5-s + (−0.0811 + 0.772i)7-s + (−0.0683 − 0.650i)9-s + (−0.749 − 1.68i)11-s + (0.0230 − 0.108i)13-s + (−0.609 + 0.198i)15-s + (0.489 − 1.09i)17-s + (0.519 − 0.110i)19-s + (0.339 − 0.305i)21-s + (0.935 + 1.28i)23-s + (−0.0934 − 0.161i)25-s + (−0.571 + 0.787i)27-s + (−1.15 − 0.375i)29-s + (−0.539 + 0.842i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0259 + 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0259 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.819525 - 0.841076i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.819525 - 0.841076i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 31 | \( 1 + (16.7 - 26.1i)T \) |
| good | 3 | \( 1 + (1.31 + 1.18i)T + (0.940 + 8.95i)T^{2} \) |
| 5 | \( 1 + (-2.72 + 4.71i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (0.568 - 5.40i)T + (-47.9 - 10.1i)T^{2} \) |
| 11 | \( 1 + (8.24 + 18.5i)T + (-80.9 + 89.9i)T^{2} \) |
| 13 | \( 1 + (-0.300 + 1.41i)T + (-154. - 68.7i)T^{2} \) |
| 17 | \( 1 + (-8.32 + 18.6i)T + (-193. - 214. i)T^{2} \) |
| 19 | \( 1 + (-9.86 + 2.09i)T + (329. - 146. i)T^{2} \) |
| 23 | \( 1 + (-21.5 - 29.6i)T + (-163. + 503. i)T^{2} \) |
| 29 | \( 1 + (33.4 + 10.8i)T + (680. + 494. i)T^{2} \) |
| 37 | \( 1 + (-34.1 + 19.7i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-37.9 - 42.1i)T + (-175. + 1.67e3i)T^{2} \) |
| 43 | \( 1 + (1.99 + 9.37i)T + (-1.68e3 + 752. i)T^{2} \) |
| 47 | \( 1 + (-1.15 - 3.56i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-65.8 + 6.92i)T + (2.74e3 - 584. i)T^{2} \) |
| 59 | \( 1 + (-15.1 + 16.8i)T + (-363. - 3.46e3i)T^{2} \) |
| 61 | \( 1 - 60.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + (31.6 - 54.7i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-1.20 - 11.4i)T + (-4.93e3 + 1.04e3i)T^{2} \) |
| 73 | \( 1 + (31.1 + 69.9i)T + (-3.56e3 + 3.96e3i)T^{2} \) |
| 79 | \( 1 + (-31.2 + 70.0i)T + (-4.17e3 - 4.63e3i)T^{2} \) |
| 83 | \( 1 + (-90.5 + 81.5i)T + (720. - 6.85e3i)T^{2} \) |
| 89 | \( 1 + (42.3 - 58.2i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (103. + 74.8i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−13.00273012521093833419446970500, −11.88865902698806379576655535043, −11.13741476672212899013972688445, −9.432396709214557502213347306901, −8.877840033677333171611037858224, −7.45372528454789964255015278140, −5.74115311759670617883857552637, −5.43800622312135194621366304606, −3.11560246130513401270547825073, −0.903643387179288126444070522808,
2.31860820732049968437442511077, 4.22195286023687504587961603587, 5.50976728023360790822800467222, 6.90088695795429384055320518983, 7.78135079530791290495188962694, 9.661682267284373188708503113947, 10.46335407652896427490735018936, 10.91961965070652895017431637090, 12.51091259276586067661907549650, 13.40195516072970956104612320453
