L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.228 + 0.395i)5-s + (2.45 − 0.989i)7-s − 0.999·8-s + 0.456·10-s + 3.83·11-s + (−3.13 − 1.78i)13-s + (0.369 − 2.61i)14-s + (−0.5 + 0.866i)16-s + (0.775 + 1.34i)17-s − 2.88·19-s + (0.228 − 0.395i)20-s + (1.91 − 3.32i)22-s + (1.62 − 2.80i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.102 + 0.176i)5-s + (0.927 − 0.374i)7-s − 0.353·8-s + 0.144·10-s + 1.15·11-s + (−0.869 − 0.494i)13-s + (0.0988 − 0.700i)14-s + (−0.125 + 0.216i)16-s + (0.188 + 0.325i)17-s − 0.661·19-s + (0.0510 − 0.0883i)20-s + (0.409 − 0.708i)22-s + (0.338 − 0.585i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0710 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0710 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.301174445\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.301174445\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.45 + 0.989i)T \) |
| 13 | \( 1 + (3.13 + 1.78i)T \) |
good | 5 | \( 1 + (-0.228 - 0.395i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 3.83T + 11T^{2} \) |
| 17 | \( 1 + (-0.775 - 1.34i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 2.88T + 19T^{2} \) |
| 23 | \( 1 + (-1.62 + 2.80i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.20 - 3.82i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.80 + 8.31i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.140 - 0.242i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.57 - 6.18i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.21 + 2.10i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.93 - 6.80i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.550 - 0.953i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.68 + 8.11i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 1.78T + 67T^{2} \) |
| 71 | \( 1 + (-5.06 + 8.77i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.40 + 2.43i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.70 - 4.69i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.35T + 83T^{2} \) |
| 89 | \( 1 + (0.0898 - 0.155i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.73 + 8.20i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.330610246373908575479007727023, −8.438592392286133739034852275019, −7.68483133269106960597400198814, −6.64710846429361038366404522644, −5.92275944662662366645000844083, −4.65652673911216309415762469041, −4.35966299371293504370156395501, −3.08314149127824459320673045548, −2.07126156185774014372992943190, −0.894450758319033610378501108969,
1.36706633605097688247346449056, 2.62103945657343960517990458527, 3.92449994618610155107089706096, 4.73759966657236023442327399564, 5.38005437329958252108723863096, 6.39744229577495683814971093473, 7.12390461215774974655400304543, 7.87437533819850901594819439302, 8.917785689823577964284531588173, 9.136713029765048025345088745721