L(s) = 1 | + (1.73 − i)5-s + (2 − 3.46i)7-s + (2.59 + 1.5i)11-s + (1.73 − i)13-s − 5·17-s − i·19-s + (−1 − 1.73i)23-s + (−0.500 + 0.866i)25-s + (−2 − 3.46i)31-s − 7.99i·35-s + 2i·37-s + (−2.5 − 4.33i)41-s + (9.52 + 5.5i)43-s + (3 − 5.19i)47-s + (−4.49 − 7.79i)49-s + ⋯ |
L(s) = 1 | + (0.774 − 0.447i)5-s + (0.755 − 1.30i)7-s + (0.783 + 0.452i)11-s + (0.480 − 0.277i)13-s − 1.21·17-s − 0.229i·19-s + (−0.208 − 0.361i)23-s + (−0.100 + 0.173i)25-s + (−0.359 − 0.622i)31-s − 1.35i·35-s + 0.328i·37-s + (−0.390 − 0.676i)41-s + (1.45 + 0.838i)43-s + (0.437 − 0.757i)47-s + (−0.642 − 1.11i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.573 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.73565 - 0.903525i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73565 - 0.903525i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.73 + i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.73 + i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 + iT - 19T^{2} \) |
| 23 | \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-9.52 - 5.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.3 - 6i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.59 - 1.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.46 - 2i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 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
−10.03611411039273972395866880190, −9.214603859292763014931941893556, −8.437600549178699437214933479142, −7.40177885448111997207301749889, −6.66454373349236651551076672880, −5.62498224518799876818552073090, −4.53713848840385478192049208780, −3.91068599888927982175961395100, −2.14138409031677671722661008055, −1.05255170127084797551357855310,
1.70344673330493652294899446059, 2.55601241655311111661337589472, 3.94013424517680362017191244686, 5.14778744650631158160840080260, 6.02853714550869518290752994316, 6.57568031628284853873208061738, 7.86387208633067262249851232596, 8.946874840541170805868643386554, 9.103157559194274156035796638571, 10.35314284608257568698468788242