| L(s) = 1 | + (0.342 + 0.939i)2-s + (−0.766 + 0.642i)4-s + (−2.57 + 0.453i)5-s + (−0.361 − 2.04i)7-s + (−0.866 − 0.500i)8-s + (−1.30 − 2.26i)10-s + (2.99 − 5.19i)11-s + (2.64 + 3.15i)13-s + (1.80 − 1.03i)14-s + (0.173 − 0.984i)16-s + (0.618 − 0.737i)17-s + (0.534 − 1.46i)19-s + (1.67 − 2.00i)20-s + (5.90 + 1.04i)22-s + (5.51 − 3.18i)23-s + ⋯ |
| L(s) = 1 | + (0.241 + 0.664i)2-s + (−0.383 + 0.321i)4-s + (−1.15 + 0.202i)5-s + (−0.136 − 0.773i)7-s + (−0.306 − 0.176i)8-s + (−0.412 − 0.715i)10-s + (0.903 − 1.56i)11-s + (0.733 + 0.874i)13-s + (0.481 − 0.277i)14-s + (0.0434 − 0.246i)16-s + (0.150 − 0.178i)17-s + (0.122 − 0.337i)19-s + (0.375 − 0.447i)20-s + (1.25 + 0.221i)22-s + (1.15 − 0.664i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.119i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.119i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.29109 - 0.0774198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29109 - 0.0774198i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.342 - 0.939i)T \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-6.07 + 0.383i)T \) |
| good | 5 | \( 1 + (2.57 - 0.453i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.361 + 2.04i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.99 + 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.64 - 3.15i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.618 + 0.737i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.534 + 1.46i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (-5.51 + 3.18i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.51 - 2.02i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.39iT - 31T^{2} \) |
| 41 | \( 1 + (7.94 - 6.66i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + 3.76iT - 43T^{2} \) |
| 47 | \( 1 + (3.08 + 5.34i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.39 + 7.90i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (5.02 + 0.885i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (6.25 + 7.45i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.83 + 10.3i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-10.1 - 3.69i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 - 3.55T + 73T^{2} \) |
| 79 | \( 1 + (-2.51 + 0.442i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (5.29 + 4.44i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-16.0 - 2.82i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (14.1 - 8.15i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.76428291104762291267928751982, −9.358290872636202746994609401756, −8.568532094079508919659968136548, −7.87423986300902971962495076634, −6.78030588507744196525647179096, −6.38414372882331104285269468053, −4.90655393075669276084180733574, −3.86873384581332430810741349790, −3.32825285484429288238902194958, −0.76215683280277522638336837626,
1.34881592513410633704648068195, 2.91814945480366620069647992350, 3.92896580894098711942845467220, 4.77260785964171383169928342979, 5.88771093609375291460271995346, 7.08173031551864704038390206210, 8.033514700084843602037112005585, 8.915080084945840783816892263320, 9.692738999890937431382216582739, 10.66692036805182079948609540685
