L(s) = 1 | + (−2.98 − 1.72i)3-s + (4.44 + 7.70i)9-s − 0.550i·11-s + (3 − 5.19i)17-s + (−2.98 − 3.17i)19-s + (−2.5 − 4.33i)25-s − 20.3i·27-s + (−0.949 + 1.64i)33-s + (−6.39 + 11.0i)41-s + (8.66 + 5i)43-s − 7·49-s + (−17.9 + 10.3i)51-s + (3.44 + 14.6i)57-s + (−8.00 − 4.62i)59-s + (−12.4 + 7.17i)67-s + ⋯ |
L(s) = 1 | + (−1.72 − 0.995i)3-s + (1.48 + 2.56i)9-s − 0.165i·11-s + (0.727 − 1.26i)17-s + (−0.685 − 0.728i)19-s + (−0.5 − 0.866i)25-s − 3.91i·27-s + (−0.165 + 0.286i)33-s + (−0.999 + 1.73i)41-s + (1.32 + 0.762i)43-s − 49-s + (−2.50 + 1.44i)51-s + (0.456 + 1.93i)57-s + (−1.04 − 0.601i)59-s + (−1.51 + 0.876i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.942 - 0.334i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2731070244\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2731070244\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.98 + 3.17i)T \) |
good | 3 | \( 1 + (2.98 + 1.72i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 0.550iT - 11T^{2} \) |
| 13 | \( 1 + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + (6.39 - 11.0i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-8.66 - 5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (8.00 + 4.62i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (12.4 - 7.17i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.84 + 13.5i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 6.55iT - 83T^{2} \) |
| 89 | \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.84 - 8.39i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.480273482482865408868167338863, −8.102445194857345015856880343019, −7.49053349076806729156958857108, −6.59525580613136668055206793102, −6.08755505641146147266960833171, −5.10950029020622880756553736639, −4.49794460619415353390529979559, −2.67580474157692121101549556324, −1.35458198694114020676825062594, −0.16169275783744718460747749585,
1.48659243628697906503626100352, 3.61968852446160698472193884090, 4.17524421400586638924160037825, 5.27806673251616090001071750173, 5.82474859363428519879145519483, 6.55763957881621352639184648609, 7.55883318971119049227188201088, 8.796379403241903452116353034504, 9.667876172253468443553995783783, 10.39048028396176387109885465542