L(s) = 1 | + (1.5 + 0.866i)5-s − 2.64·7-s + (−3.96 + 2.29i)11-s − 3.46i·13-s + (−4.5 + 2.59i)17-s + (−1.32 + 2.29i)19-s + (−3.96 − 2.29i)23-s + (−1 − 1.73i)25-s + (1.32 + 2.29i)31-s + (−3.96 − 2.29i)35-s + (−3.5 + 6.06i)37-s − 3.46i·41-s − 9.16i·43-s + (−3.96 + 6.87i)47-s + 7.00·49-s + ⋯ |
L(s) = 1 | + (0.670 + 0.387i)5-s − 0.999·7-s + (−1.19 + 0.690i)11-s − 0.960i·13-s + (−1.09 + 0.630i)17-s + (−0.303 + 0.525i)19-s + (−0.827 − 0.477i)23-s + (−0.200 − 0.346i)25-s + (0.237 + 0.411i)31-s + (−0.670 − 0.387i)35-s + (−0.575 + 0.996i)37-s − 0.541i·41-s − 1.39i·43-s + (−0.578 + 1.00i)47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2476878935\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2476878935\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 2.64T \) |
good | 5 | \( 1 + (-1.5 - 0.866i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.96 - 2.29i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + (4.5 - 2.59i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.32 - 2.29i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.96 + 2.29i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-1.32 - 2.29i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.5 - 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.46iT - 41T^{2} \) |
| 43 | \( 1 + 9.16iT - 43T^{2} \) |
| 47 | \( 1 + (3.96 - 6.87i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.96 + 6.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.5 - 0.866i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.96 - 2.29i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.16iT - 71T^{2} \) |
| 73 | \( 1 + (4.5 - 2.59i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.96 - 2.29i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (1.5 + 0.866i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3.46iT - 97T^{2} \) |
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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
−10.28279798689886151661312613252, −9.842320007172630554727307762236, −8.682047818553468861982044978363, −7.901360579886362904830349292478, −6.84304158859718711240011749337, −6.15918770743544933030043835052, −5.33877641167049481110865710861, −4.13955444668826971207997872736, −2.92123545981488831272916035124, −2.09701923026291125310806339808,
0.10050117155282918327110425986, 2.02112644427958919218154372903, 3.01945934665557814152385975183, 4.27666683240043686262635125860, 5.31554391104544217797693165917, 6.13501858343338108528619396526, 6.89873427578238172091999417391, 7.938168923503822329196980092494, 9.012940866819939865859100205414, 9.423801890902206756870536684151