| L(s) = 1 | + (−1.07 + 2.10i)2-s + (0.270 − 1.71i)3-s + (−0.933 − 1.28i)4-s + (4.10 − 2.85i)5-s + (3.31 + 2.40i)6-s + (9.10 + 9.10i)7-s + (−5.63 + 0.891i)8-s + (−2.85 − 0.927i)9-s + (1.61 + 11.7i)10-s + (−0.412 − 1.26i)11-s + (−2.45 + 1.24i)12-s + (2.93 + 5.75i)13-s + (−28.9 + 9.40i)14-s + (−3.77 − 7.79i)15-s + (6.12 − 18.8i)16-s + (0.410 + 2.59i)17-s + ⋯ |
| L(s) = 1 | + (−0.536 + 1.05i)2-s + (0.0903 − 0.570i)3-s + (−0.233 − 0.321i)4-s + (0.820 − 0.571i)5-s + (0.552 + 0.401i)6-s + (1.30 + 1.30i)7-s + (−0.703 + 0.111i)8-s + (−0.317 − 0.103i)9-s + (0.161 + 1.17i)10-s + (−0.0374 − 0.115i)11-s + (−0.204 + 0.104i)12-s + (0.225 + 0.442i)13-s + (−2.06 + 0.671i)14-s + (−0.251 − 0.519i)15-s + (0.382 − 1.17i)16-s + (0.0241 + 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.434 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.01009 + 0.634363i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01009 + 0.634363i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.270 + 1.71i)T \) |
| 5 | \( 1 + (-4.10 + 2.85i)T \) |
| good | 2 | \( 1 + (1.07 - 2.10i)T + (-2.35 - 3.23i)T^{2} \) |
| 7 | \( 1 + (-9.10 - 9.10i)T + 49iT^{2} \) |
| 11 | \( 1 + (0.412 + 1.26i)T + (-97.8 + 71.1i)T^{2} \) |
| 13 | \( 1 + (-2.93 - 5.75i)T + (-99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (-0.410 - 2.59i)T + (-274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-3.46 + 4.77i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (-6.24 - 3.17i)T + (310. + 427. i)T^{2} \) |
| 29 | \( 1 + (33.9 + 46.6i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (29.8 + 21.6i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (47.6 - 24.2i)T + (804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (4.11 - 12.6i)T + (-1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + (-40.0 + 40.0i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (48.6 + 7.70i)T + (2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (8.18 - 51.6i)T + (-2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (-61.4 - 19.9i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (4.60 + 14.1i)T + (-3.01e3 + 2.18e3i)T^{2} \) |
| 67 | \( 1 + (13.8 + 87.6i)T + (-4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-12.8 + 9.30i)T + (1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (96.0 + 48.9i)T + (3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-62.5 - 86.0i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-102. + 16.2i)T + (6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (83.2 - 27.0i)T + (6.40e3 - 4.65e3i)T^{2} \) |
| 97 | \( 1 + (41.9 + 6.65i)T + (8.94e3 + 2.90e3i)T^{2} \) |
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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
−14.71469231932309597634666048232, −13.62200223280849674129319473310, −12.30270861691605779301879462607, −11.40564796516036760466436499585, −9.318642234265125000178963678447, −8.632459869341511192201391067845, −7.67818570592782908614998275516, −6.13901944031576724683971658193, −5.28502169111531094893300874095, −2.07118130643992400731884144346,
1.62843724016173936331091177775, 3.49236818600879459957363477158, 5.32492401035152684236306457821, 7.20063583669723800552838542368, 8.783902803947834241748881916673, 10.01165770875007360717696638571, 10.75293343365379461865675665933, 11.24779816715497087280562494068, 12.92736837712838386057154904225, 14.27659727920941351268676649174
