L(s) = 1 | + 0.512·2-s − 3.19i·3-s − 1.73·4-s − 5-s − 1.64i·6-s + 1.77i·7-s − 1.91·8-s − 7.23·9-s − 0.512·10-s + (−0.881 + 3.19i)11-s + 5.55i·12-s + 2.41·13-s + 0.909i·14-s + 3.19i·15-s + 2.49·16-s − 1.90i·17-s + ⋯ |
L(s) = 1 | + 0.362·2-s − 1.84i·3-s − 0.868·4-s − 0.447·5-s − 0.669i·6-s + 0.669i·7-s − 0.677·8-s − 2.41·9-s − 0.162·10-s + (−0.265 + 0.964i)11-s + 1.60i·12-s + 0.669·13-s + 0.242i·14-s + 0.825i·15-s + 0.622·16-s − 0.462i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6365569633\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6365569633\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + T \) |
| 11 | \( 1 + (0.881 - 3.19i)T \) |
| 19 | \( 1 + (-3.68 - 2.33i)T \) |
good | 2 | \( 1 - 0.512T + 2T^{2} \) |
| 3 | \( 1 + 3.19iT - 3T^{2} \) |
| 7 | \( 1 - 1.77iT - 7T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 + 1.90iT - 17T^{2} \) |
| 23 | \( 1 + 3.67T + 23T^{2} \) |
| 29 | \( 1 + 7.99T + 29T^{2} \) |
| 31 | \( 1 + 0.955iT - 31T^{2} \) |
| 37 | \( 1 - 6.90iT - 37T^{2} \) |
| 41 | \( 1 - 0.482T + 41T^{2} \) |
| 43 | \( 1 - 12.0iT - 43T^{2} \) |
| 47 | \( 1 - 0.519T + 47T^{2} \) |
| 53 | \( 1 - 2.92iT - 53T^{2} \) |
| 59 | \( 1 + 10.0iT - 59T^{2} \) |
| 61 | \( 1 + 7.83iT - 61T^{2} \) |
| 67 | \( 1 - 3.84iT - 67T^{2} \) |
| 71 | \( 1 - 0.976iT - 71T^{2} \) |
| 73 | \( 1 - 0.0771iT - 73T^{2} \) |
| 79 | \( 1 + 7.50T + 79T^{2} \) |
| 83 | \( 1 - 2.81iT - 83T^{2} \) |
| 89 | \( 1 - 15.4iT - 89T^{2} \) |
| 97 | \( 1 - 6.02iT - 97T^{2} \) |
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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
−9.771162472968351537926891504575, −9.013864517825326594874633530840, −8.012988782119217314941348423605, −7.73947003978978580138432220011, −6.61104208826699122779090557275, −5.81619761219185476012838203827, −5.02838260423712016433542107150, −3.65677923059828225333086061125, −2.53910746759556114445787090979, −1.29556183000666905987768579678,
0.28489012145156323822903982313, 3.14566288217542626285494916283, 3.85293555615216729356235422503, 4.24617293483051347976437838068, 5.43075143650850500668739440409, 5.80185463090565053270647310654, 7.51027833374781958132120901781, 8.632596219884299744357253357437, 8.933486251620699879571419602775, 9.892615833089893706751486759936