L(s) = 1 | + (−0.184 − 0.184i)2-s + (−0.866 − 0.5i)3-s − 1.93i·4-s + (0.900 + 3.35i)5-s + (0.0673 + 0.251i)6-s + (0.939 + 2.47i)7-s + (−0.723 + 0.723i)8-s + (0.499 + 0.866i)9-s + (0.452 − 0.784i)10-s + (0.621 + 2.31i)11-s + (−0.966 + 1.67i)12-s + (1.60 − 3.23i)13-s + (0.282 − 0.628i)14-s + (0.900 − 3.35i)15-s − 3.59·16-s + 3.34·17-s + ⋯ |
L(s) = 1 | + (−0.130 − 0.130i)2-s + (−0.499 − 0.288i)3-s − 0.966i·4-s + (0.402 + 1.50i)5-s + (0.0275 + 0.102i)6-s + (0.355 + 0.934i)7-s + (−0.255 + 0.255i)8-s + (0.166 + 0.288i)9-s + (0.143 − 0.247i)10-s + (0.187 + 0.699i)11-s + (−0.278 + 0.483i)12-s + (0.443 − 0.896i)13-s + (0.0754 − 0.167i)14-s + (0.232 − 0.867i)15-s − 0.899·16-s + 0.810·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11659 + 0.190902i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11659 + 0.190902i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (-0.939 - 2.47i)T \) |
| 13 | \( 1 + (-1.60 + 3.23i)T \) |
good | 2 | \( 1 + (0.184 + 0.184i)T + 2iT^{2} \) |
| 5 | \( 1 + (-0.900 - 3.35i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-0.621 - 2.31i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 - 3.34T + 17T^{2} \) |
| 19 | \( 1 + (-6.06 - 1.62i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + 0.157iT - 23T^{2} \) |
| 29 | \( 1 + (-0.530 - 0.919i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (5.14 + 1.37i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (1.73 - 1.73i)T - 37iT^{2} \) |
| 41 | \( 1 + (-11.0 - 2.95i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (6.74 + 3.89i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (7.60 - 2.03i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (3.19 + 5.54i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.54 + 2.54i)T + 59iT^{2} \) |
| 61 | \( 1 + (-10.8 + 6.24i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.12 - 1.10i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-3.09 + 0.829i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-1.97 + 7.36i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-3.17 + 5.49i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (10.4 - 10.4i)T - 83iT^{2} \) |
| 89 | \( 1 + (10.1 + 10.1i)T + 89iT^{2} \) |
| 97 | \( 1 + (-1.40 - 5.24i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−11.64215571122744021299534561265, −11.03082855959801856317413385666, −10.11251460472674732469773832176, −9.519286887798173896176456846267, −7.918824590405296999776811737407, −6.82728476358971756305143794733, −5.88138731026475609450241413777, −5.24946413558737575842514223348, −3.06990022026645824920717759881, −1.72861415946057064980432245562,
1.13161976464362083863816301562, 3.62007132419139238613987262452, 4.59153004226418047367082116438, 5.64533362025442676570253229616, 7.04353073761006560635146809725, 8.073050815062081295498626255057, 8.991651709662140649023444051279, 9.712214513532698214279726515462, 11.16214004442566304602298271761, 11.81876293666222328530744325155