L(s) = 1 | − 2.41·2-s + (1 + 1.41i)3-s + 3.82·4-s + i·5-s + (−2.41 − 3.41i)6-s + i·7-s − 4.41·8-s + (−1.00 + 2.82i)9-s − 2.41i·10-s + (−1.41 + 3i)11-s + (3.82 + 5.41i)12-s − 2.41i·14-s + (−1.41 + i)15-s + 2.99·16-s + 1.17·17-s + (2.41 − 6.82i)18-s + ⋯ |
L(s) = 1 | − 1.70·2-s + (0.577 + 0.816i)3-s + 1.91·4-s + 0.447i·5-s + (−0.985 − 1.39i)6-s + 0.377i·7-s − 1.56·8-s + (−0.333 + 0.942i)9-s − 0.763i·10-s + (−0.426 + 0.904i)11-s + (1.10 + 1.56i)12-s − 0.645i·14-s + (−0.365 + 0.258i)15-s + 0.749·16-s + 0.284·17-s + (0.569 − 1.60i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.174i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5055815062\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5055815062\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1 - 1.41i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + (1.41 - 3i)T \) |
good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 19 | \( 1 - 4.82iT - 19T^{2} \) |
| 23 | \( 1 + 4.82iT - 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 + 4.82T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 1.65iT - 43T^{2} \) |
| 47 | \( 1 - 5.17iT - 47T^{2} \) |
| 53 | \( 1 + 4.82iT - 53T^{2} \) |
| 59 | \( 1 + 11.6iT - 59T^{2} \) |
| 61 | \( 1 - 10.8iT - 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 - 11.6iT - 71T^{2} \) |
| 73 | \( 1 + 12iT - 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + 3.65iT - 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−9.890564576861570354257543392385, −9.592297786265952574115518747831, −8.678061624282053819488454305874, −7.956309482668748300691098259775, −7.41658315979107512340847711271, −6.33146585074858009467525168055, −5.18326799711102433496251762636, −3.89559839357338646144793355676, −2.65901321477146652979215573245, −1.85544647224117730049348437037,
0.35891948470655134718687340536, 1.39632213943218962268505421934, 2.51326560655555123499681572560, 3.64930127054741408285637396511, 5.40073527479789986487080806539, 6.44812847748733345634131191308, 7.35065036165764919889751796892, 7.80366188763821810109898663724, 8.600983412929338630313211793170, 9.192023172077861770386953368732