L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 0.717·5-s + (−1 + 2.44i)7-s + 0.999i·8-s + (−0.621 + 0.358i)10-s − 3i·11-s + (−2.12 + 1.22i)13-s + (−0.358 − 2.62i)14-s + (−0.5 − 0.866i)16-s + (−2.95 − 5.12i)17-s + (−5.12 − 2.95i)19-s + (0.358 − 0.621i)20-s + (1.5 + 2.59i)22-s − 4.24i·23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.320·5-s + (−0.377 + 0.925i)7-s + 0.353i·8-s + (−0.196 + 0.113i)10-s − 0.904i·11-s + (−0.588 + 0.339i)13-s + (−0.0958 − 0.700i)14-s + (−0.125 − 0.216i)16-s + (−0.717 − 1.24i)17-s + (−1.17 − 0.678i)19-s + (0.0802 − 0.138i)20-s + (0.319 + 0.553i)22-s − 0.884i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.164 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5257243583\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5257243583\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
good | 5 | \( 1 - 0.717T + 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + (2.12 - 1.22i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.95 + 5.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.12 + 2.95i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 4.24iT - 23T^{2} \) |
| 29 | \( 1 + (-6.27 - 3.62i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.86 - 4.54i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.121 + 0.210i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.91 + 10.2i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.121 + 0.210i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.95 + 5.12i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.27 - 3.62i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.03 + 6.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.878 - 0.507i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5 + 8.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.75iT - 71T^{2} \) |
| 73 | \( 1 + (1.24 - 0.717i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.37 - 2.38i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.31 - 5.74i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.19 + 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.7 + 6.77i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.434512592754558694326314958960, −8.718419531054014985415692377816, −8.281073866736655176361210537733, −6.78340122220352737664518900037, −6.56286097296670593747894600963, −5.42819754494246860381687939213, −4.62286230174403243802044471760, −2.95465489508182444121239513234, −2.15510393218552044952009364451, −0.27073879061932702441261267532,
1.49267475292088127285024808914, 2.56486668140189334861380391947, 3.90934302797270181849912332198, 4.60010176220807330890005509563, 6.15914198571384437962179586482, 6.69522450588144140158005395154, 7.84215435418683274727579674155, 8.244989233461765659888967277889, 9.578091997793888254683132647576, 10.01003916749249612785844283554