| L(s) = 1 | + (0.173 + 0.984i)2-s + (−0.939 + 0.342i)4-s + (1.79 + 1.50i)5-s + (1.93 + 1.62i)7-s + (−0.5 − 0.866i)8-s + (−1.17 + 2.03i)10-s + (0.560 + 0.970i)11-s + (1.70 − 0.620i)13-s + (−1.26 + 2.19i)14-s + (0.766 − 0.642i)16-s + (−2.81 − 1.02i)17-s + (−0.971 + 5.51i)19-s + (−2.20 − 0.802i)20-s + (−0.858 + 0.720i)22-s + (4.55 − 7.88i)23-s + ⋯ |
| L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.469 + 0.171i)4-s + (0.804 + 0.674i)5-s + (0.733 + 0.615i)7-s + (−0.176 − 0.306i)8-s + (−0.371 + 0.642i)10-s + (0.168 + 0.292i)11-s + (0.473 − 0.172i)13-s + (−0.338 + 0.586i)14-s + (0.191 − 0.160i)16-s + (−0.683 − 0.248i)17-s + (−0.222 + 1.26i)19-s + (−0.493 − 0.179i)20-s + (−0.183 + 0.153i)22-s + (0.949 − 1.64i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12743 + 1.45992i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12743 + 1.45992i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-2.33 - 5.61i)T \) |
| good | 5 | \( 1 + (-1.79 - 1.50i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.93 - 1.62i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.560 - 0.970i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.70 + 0.620i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (2.81 + 1.02i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (0.971 - 5.51i)T + (-17.8 - 6.49i)T^{2} \) |
| 23 | \( 1 + (-4.55 + 7.88i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.52 - 7.83i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4.53T + 31T^{2} \) |
| 41 | \( 1 + (6.98 - 2.54i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + 8.92T + 43T^{2} \) |
| 47 | \( 1 + (0.194 - 0.337i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.23 + 7.74i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (6.41 - 5.38i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-2.45 + 0.892i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-5.62 - 4.72i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.448 + 2.54i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + 0.709T + 73T^{2} \) |
| 79 | \( 1 + (-3.39 - 2.84i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (6.81 + 2.47i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-9.95 + 8.35i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-7.34 + 12.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.58524451745179144437629201386, −9.965670543874462217622491060731, −8.695736606027262494121084033041, −8.375275917146387195262823608814, −6.95277303092585468004111649302, −6.43331797082258467012921138741, −5.42818348129997615917702026707, −4.58403895264579036840563196224, −3.10760652144914542197810080933, −1.82046391257734116991192025389,
1.06352130561113219681091042686, 2.13816211043781266132257226173, 3.66123128764100168503709583198, 4.71018018271171822501779513174, 5.45440563984686329177594010423, 6.61594033679947171394742028763, 7.79222681921137559787204418487, 8.902067932933833724908497717581, 9.303327758041657194838829575407, 10.39638555333584920010182324625
