L(s) = 1 | + 2-s + 4-s + (−1.5 + 2.59i)5-s + 8-s + (−1.5 + 2.59i)10-s + (−1.5 − 2.59i)11-s + (−0.5 − 0.866i)13-s + 16-s + (−1.5 + 2.59i)17-s + (−3.5 − 6.06i)19-s + (−1.5 + 2.59i)20-s + (−1.5 − 2.59i)22-s + (−4.5 + 7.79i)23-s + (−2 − 3.46i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.670 + 1.16i)5-s + 0.353·8-s + (−0.474 + 0.821i)10-s + (−0.452 − 0.783i)11-s + (−0.138 − 0.240i)13-s + 0.250·16-s + (−0.363 + 0.630i)17-s + (−0.802 − 1.39i)19-s + (−0.335 + 0.580i)20-s + (−0.319 − 0.553i)22-s + (−0.938 + 1.62i)23-s + (−0.400 − 0.692i)25-s + (−0.0980 − 0.169i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.5 - 7.79i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.5 + 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (-1.5 + 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.5 + 2.59i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.350162035565962120298005670892, −7.55713329745651211248237912120, −7.03143017555354386618551610216, −6.16543752610544978727109387986, −5.52500386476360807129813953531, −4.42394985463542690387726380921, −3.60481960106364340740353551097, −2.98134475442050176936067560392, −1.99442414545846189084049874956, 0,
1.59850526679908483794222324322, 2.58578685073989810568903193098, 3.94494665523605340093394526127, 4.39746723970510222075624008082, 5.09223055653503548243724302807, 5.95738229707246535965970626520, 6.87852161832390512043012319430, 7.68732714419464148488953201406, 8.350799668777998441207730296158