L(s) = 1 | + (−0.681 − 2.54i)2-s + (−0.965 − 0.258i)3-s + (−4.26 + 2.46i)4-s + (−2.18 + 0.478i)5-s + 2.63i·6-s + (5.45 + 5.45i)8-s + (0.866 + 0.499i)9-s + (2.70 + 5.22i)10-s + (0.731 + 1.26i)11-s + (4.76 − 1.27i)12-s + (−0.887 + 0.887i)13-s + (2.23 + 0.103i)15-s + (5.22 − 9.04i)16-s + (0.770 − 2.87i)17-s + (0.681 − 2.54i)18-s + (−1.97 + 3.42i)19-s + ⋯ |
L(s) = 1 | + (−0.481 − 1.79i)2-s + (−0.557 − 0.149i)3-s + (−2.13 + 1.23i)4-s + (−0.976 + 0.213i)5-s + 1.07i·6-s + (1.92 + 1.92i)8-s + (0.288 + 0.166i)9-s + (0.855 + 1.65i)10-s + (0.220 + 0.381i)11-s + (1.37 − 0.368i)12-s + (−0.246 + 0.246i)13-s + (0.576 + 0.0267i)15-s + (1.30 − 2.26i)16-s + (0.186 − 0.697i)17-s + (0.160 − 0.599i)18-s + (−0.454 + 0.786i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.292030 - 0.513458i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.292030 - 0.513458i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 + (2.18 - 0.478i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.681 + 2.54i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (-0.731 - 1.26i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.887 - 0.887i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.770 + 2.87i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.97 - 3.42i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.64 + 1.51i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 5.18iT - 29T^{2} \) |
| 31 | \( 1 + (5.28 - 3.05i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.825 - 3.08i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 0.769iT - 41T^{2} \) |
| 43 | \( 1 + (5.18 + 5.18i)T + 43iT^{2} \) |
| 47 | \( 1 + (-11.7 + 3.13i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.199 - 0.743i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (1.59 + 2.76i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.23 + 0.710i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.10 - 2.17i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 7.62T + 71T^{2} \) |
| 73 | \( 1 + (-9.30 - 2.49i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.91 - 2.26i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.75 + 6.75i)T - 83iT^{2} \) |
| 89 | \( 1 + (-0.599 + 1.03i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.68 - 8.68i)T + 97iT^{2} \) |
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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
−10.37941078519757666788976130074, −9.504346798101598635434332531567, −8.651867909649730382028980685012, −7.74006931630986253397400599732, −6.82402077443530510525081212299, −5.11845077036916619035278626238, −4.23400561591974228275620170968, −3.36128578620864624421919994052, −2.13649980977366905223447193527, −0.67675534729583733062235122390,
0.74721852781424225259354661273, 3.69017817028017394209479924662, 4.73955295323397279001741569169, 5.43705033121786655291731905497, 6.44752806455078975984765530094, 7.20218769007066744735598848090, 7.892883998450100542455724279836, 8.812990711923029458806890270724, 9.337223339653441817688349008564, 10.59948622664722261547061799914