| L(s) = 1 | + (0.304 − 0.0481i)3-s + (−1.19 + 4.85i)5-s + (5.20 + 5.20i)7-s + (−8.46 + 2.75i)9-s + (−0.00492 + 0.0151i)11-s + (−7.35 − 3.74i)13-s + (−0.128 + 1.53i)15-s + (−16.3 − 2.59i)17-s + (1.93 + 2.65i)19-s + (1.83 + 1.33i)21-s + (−5.56 − 10.9i)23-s + (−22.1 − 11.5i)25-s + (−4.91 + 2.50i)27-s + (−22.5 + 31.0i)29-s + (19.1 − 13.9i)31-s + ⋯ |
| L(s) = 1 | + (0.101 − 0.0160i)3-s + (−0.238 + 0.971i)5-s + (0.744 + 0.744i)7-s + (−0.941 + 0.305i)9-s + (−0.000447 + 0.00137i)11-s + (−0.566 − 0.288i)13-s + (−0.00857 + 0.102i)15-s + (−0.962 − 0.152i)17-s + (0.101 + 0.139i)19-s + (0.0873 + 0.0634i)21-s + (−0.242 − 0.475i)23-s + (−0.886 − 0.462i)25-s + (−0.181 + 0.0926i)27-s + (−0.778 + 1.07i)29-s + (0.618 − 0.449i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.247421 + 0.886738i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.247421 + 0.886738i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.19 - 4.85i)T \) |
| good | 3 | \( 1 + (-0.304 + 0.0481i)T + (8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (-5.20 - 5.20i)T + 49iT^{2} \) |
| 11 | \( 1 + (0.00492 - 0.0151i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (7.35 + 3.74i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (16.3 + 2.59i)T + (274. + 89.3i)T^{2} \) |
| 19 | \( 1 + (-1.93 - 2.65i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (5.56 + 10.9i)T + (-310. + 427. i)T^{2} \) |
| 29 | \( 1 + (22.5 - 31.0i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-19.1 + 13.9i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (7.24 - 14.2i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-12.8 - 39.4i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (46.2 - 46.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-4.56 - 28.8i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-38.5 + 6.10i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (44.3 - 14.4i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (5.55 - 17.0i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (-82.3 - 13.0i)T + (4.26e3 + 1.38e3i)T^{2} \) |
| 71 | \( 1 + (87.7 + 63.7i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-56.9 - 111. i)T + (-3.13e3 + 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-78.7 + 108. i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (20.4 - 129. i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + (65.2 + 21.2i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (10.8 + 68.4i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−11.37262324639641856693351128326, −10.75661328201473016843052418741, −9.621420469455020968807927321158, −8.521211587663303275798070561692, −7.86170403718783570420595487410, −6.73452600968726707650967645875, −5.70423036740537124604655552362, −4.64657813229911366665468390801, −3.08007320464387799960869046710, −2.19027578309555907928008602685,
0.37172999348267388042135755697, 2.01519166862722842422111901266, 3.78275980848358430726086472732, 4.71676771091275469447369374241, 5.69377464501441418806429458147, 7.05919589451368723680975625244, 8.039029009983094753006813661546, 8.770865816174370621529961357101, 9.624441676823252360927656697929, 10.83764317035942976124365238631
