L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.18 + 1.26i)3-s + (−0.499 + 0.866i)4-s + 0.792i·5-s + (0.5 − 1.65i)6-s + (0.5 + 2.59i)7-s + 0.999·8-s + (−0.186 + 2.99i)9-s + (0.686 − 0.396i)10-s + (−2.18 − 3.78i)11-s + (−1.68 + 0.396i)12-s + (3.5 + 0.866i)13-s + (2 − 1.73i)14-s + (−1 + 0.939i)15-s + (−0.5 − 0.866i)16-s + (−2.18 + 3.78i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.684 + 0.728i)3-s + (−0.249 + 0.433i)4-s + 0.354i·5-s + (0.204 − 0.677i)6-s + (0.188 + 0.981i)7-s + 0.353·8-s + (−0.0620 + 0.998i)9-s + (0.216 − 0.125i)10-s + (−0.659 − 1.14i)11-s + (−0.486 + 0.114i)12-s + (0.970 + 0.240i)13-s + (0.534 − 0.462i)14-s + (−0.258 + 0.242i)15-s + (−0.125 − 0.216i)16-s + (−0.530 + 0.918i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12673 + 0.781381i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12673 + 0.781381i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.18 - 1.26i)T \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
| 13 | \( 1 + (-3.5 - 0.866i)T \) |
good | 5 | \( 1 - 0.792iT - 5T^{2} \) |
| 11 | \( 1 + (2.18 + 3.78i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2.18 - 3.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.18 - 2.05i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.68 - 2.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.18 - 1.26i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 + (10.1 - 5.84i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.18 + 4.72i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 0.939iT - 47T^{2} \) |
| 53 | \( 1 - 2.22iT - 53T^{2} \) |
| 59 | \( 1 + (-5.31 - 3.06i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.1 + 5.84i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.05 + 13.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 9.62T + 79T^{2} \) |
| 83 | \( 1 + 1.58iT - 83T^{2} \) |
| 89 | \( 1 + (-9.30 + 5.37i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.372 - 0.644i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.76159431409444881683089407613, −10.29179730426089407640361849458, −9.033566501119402696354490860026, −8.570726541757673816403099336665, −7.938013502616224845201363177331, −6.29145799002460928482155558127, −5.28337723907797744907851242989, −3.92737892537179869299343608314, −3.07639802422447340172642941994, −1.98555108611611931647074163913,
0.851344135118495632976285358205, 2.35093333252676357519013681536, 3.99720999232823917148758877441, 5.01138476228932580053460010461, 6.45818511371480079272192802099, 7.13599096223703272565421893644, 7.911867991805647015733996704220, 8.635769782844453361836185708342, 9.564817606304717243472454258847, 10.42226150759987397819689454546