L(s) = 1 | + (−0.837 − 1.81i)2-s + (−1.32 − 2.69i)3-s + (−2.59 + 3.04i)4-s + (−3.78 + 4.65i)6-s + (3.54 + 3.54i)7-s + (7.70 + 2.16i)8-s + (−5.50 + 7.12i)9-s + 16.8·11-s + (11.6 + 2.96i)12-s + (8.64 + 8.64i)13-s + (3.46 − 9.40i)14-s + (−2.51 − 15.8i)16-s + (−9.72 − 9.72i)17-s + (17.5 + 4.03i)18-s + 4.78·19-s + ⋯ |
L(s) = 1 | + (−0.418 − 0.908i)2-s + (−0.440 − 0.897i)3-s + (−0.649 + 0.760i)4-s + (−0.630 + 0.776i)6-s + (0.506 + 0.506i)7-s + (0.962 + 0.270i)8-s + (−0.611 + 0.791i)9-s + 1.53·11-s + (0.968 + 0.247i)12-s + (0.665 + 0.665i)13-s + (0.247 − 0.671i)14-s + (−0.157 − 0.987i)16-s + (−0.572 − 0.572i)17-s + (0.974 + 0.224i)18-s + 0.251·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 + 0.954i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.964424 - 0.708584i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.964424 - 0.708584i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.837 + 1.81i)T \) |
| 3 | \( 1 + (1.32 + 2.69i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-3.54 - 3.54i)T + 49iT^{2} \) |
| 11 | \( 1 - 16.8T + 121T^{2} \) |
| 13 | \( 1 + (-8.64 - 8.64i)T + 169iT^{2} \) |
| 17 | \( 1 + (9.72 + 9.72i)T + 289iT^{2} \) |
| 19 | \( 1 - 4.78T + 361T^{2} \) |
| 23 | \( 1 + (-13.5 - 13.5i)T + 529iT^{2} \) |
| 29 | \( 1 + 14.8T + 841T^{2} \) |
| 31 | \( 1 - 14.0iT - 961T^{2} \) |
| 37 | \( 1 + (-10.1 + 10.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 6.08iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-57.2 + 57.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-17.6 + 17.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (16.2 - 16.2i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 4.37iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 8.52T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-53.9 - 53.9i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 36.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-12.6 - 12.6i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 88.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + (63.7 + 63.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 115.T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-85.3 + 85.3i)T - 9.40e3iT^{2} \) |
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
−11.61501363455863306997511079970, −10.75586733812347697393440497942, −9.206593278679864775159066411859, −8.788997601697308161573818022309, −7.52806456197483351902627799242, −6.57765867161470322162656370401, −5.21132283158411631587066290945, −3.84978536029046854948350472372, −2.19377850847927078791768848315, −1.10783458291915389109961568352,
0.999296483847152313892065710061, 3.83425674698506130917997392379, 4.64924280398402240178481393984, 5.90521238825383873551267279975, 6.64503242314253757809974586870, 7.964637206366548308715558204855, 8.951546733009801625283425180681, 9.628167463778659474883041852436, 10.79367739660869433715002618011, 11.22127379418078667164085849287