| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0729i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.997 + 0.0729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00733642 - 0.200995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00733642 - 0.200995i\) |
| \(L(2)\) |
|
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 | \( 1 + (10.8 - 68.4i)T + (-8.94e3 - 2.90e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.93939153302740012116477996353, −10.95470246089658062637346476565, −9.353762825236376667289484277032, −9.022097621800945556141112928762, −7.82464418911843317897903192131, −6.33915338868964645215775952392, −5.43852668794009785492580078167, −4.08959913486899005188136953765, −2.44599573263809673502476040722, −0.10724356276492724246391348577,
2.64200795210659981918030806870, 3.78395260623664929361975703728, 5.41067756375675006053527299217, 6.71603465094559012048805539780, 7.39504573417435496208187361243, 8.741590001716286477936983941329, 9.947204515334194290816854696482, 10.82869020108638281653411471113, 11.48572612791677863978489679403, 12.77139478224694893550414120203
