| L(s) = 1 | + (−0.642 + 1.26i)2-s + (−0.270 + 1.71i)3-s + (−1.17 − 1.61i)4-s + (−2.24 + 4.46i)5-s + (−1.98 − 1.43i)6-s + (−8.57 − 8.57i)7-s + (2.79 − 0.442i)8-s + (−2.85 − 0.927i)9-s + (−4.18 − 5.69i)10-s + (0.681 + 2.09i)11-s + (3.08 − 1.57i)12-s + (−0.642 − 1.26i)13-s + (16.3 − 5.30i)14-s + (−7.03 − 5.05i)15-s + (−1.23 + 3.80i)16-s + (−3.13 − 19.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.321 + 0.630i)2-s + (−0.0903 + 0.570i)3-s + (−0.293 − 0.404i)4-s + (−0.449 + 0.893i)5-s + (−0.330 − 0.239i)6-s + (−1.22 − 1.22i)7-s + (0.349 − 0.0553i)8-s + (−0.317 − 0.103i)9-s + (−0.418 − 0.569i)10-s + (0.0619 + 0.190i)11-s + (0.257 − 0.131i)12-s + (−0.0494 − 0.0969i)13-s + (1.16 − 0.378i)14-s + (−0.468 − 0.336i)15-s + (−0.0772 + 0.237i)16-s + (−0.184 − 1.16i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0391952 - 0.0704440i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0391952 - 0.0704440i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 - 1.26i)T \) |
| 3 | \( 1 + (0.270 - 1.71i)T \) |
| 5 | \( 1 + (2.24 - 4.46i)T \) |
| good | 7 | \( 1 + (8.57 + 8.57i)T + 49iT^{2} \) |
| 11 | \( 1 + (-0.681 - 2.09i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (0.642 + 1.26i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (3.13 + 19.7i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (14.3 - 19.7i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (21.3 + 10.8i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (-22.2 - 30.5i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (23.8 + 17.3i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (7.25 - 3.69i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (21.4 - 66.0i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (26.5 - 26.5i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (16.5 + 2.62i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (12.8 - 81.3i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-27.3 - 8.87i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (20.9 + 64.5i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (1.94 + 12.2i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-97.9 + 71.1i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-91.3 - 46.5i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (66.9 + 92.1i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (78.3 - 12.4i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (0.353 - 0.114i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (86.8 + 13.7i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−13.77203028634661946513955813371, −12.48853394644022143591528803484, −11.07161855915661609055296321528, −10.20686344897159226493447781080, −9.625348808805223683913142366250, −8.062669170527813982235619622270, −6.98578225342290777074209245167, −6.25262695480759975650578689633, −4.42264958819874127346180811318, −3.28351207556898226771520406338,
0.05511945933139242973575086440, 2.15572870001148366073556301974, 3.76130156729666560227485937279, 5.48757686007596681307284545907, 6.69735553297371403286526489470, 8.331304357952991955056482463783, 8.871629545752914034058662426982, 9.961259248935907959272792243075, 11.38026015326546110254108190575, 12.29109845773352126167756416491
