L(s) = 1 | − 2-s + 4-s + (1.5 + 2.59i)5-s − 8-s + (−1.5 − 2.59i)10-s + (−1.5 + 2.59i)11-s + (2.5 − 4.33i)13-s + 16-s + (−1.5 − 2.59i)17-s + (2.5 − 4.33i)19-s + (1.5 + 2.59i)20-s + (1.5 − 2.59i)22-s + (−1.5 − 2.59i)23-s + (−2 + 3.46i)25-s + (−2.5 + 4.33i)26-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.670 + 1.16i)5-s − 0.353·8-s + (−0.474 − 0.821i)10-s + (−0.452 + 0.783i)11-s + (0.693 − 1.20i)13-s + 0.250·16-s + (−0.363 − 0.630i)17-s + (0.573 − 0.993i)19-s + (0.335 + 0.580i)20-s + (0.319 − 0.553i)22-s + (−0.312 − 0.541i)23-s + (−0.400 + 0.692i)25-s + (−0.490 + 0.849i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.350025938\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.350025938\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.5 + 4.33i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + (-3.5 + 6.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (1.5 + 2.59i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.868667717466868427297937666111, −7.997601305115960714750129300333, −7.23308592188714657938947376942, −6.73394335083558953687109286973, −5.85218606543768584543169184627, −5.11846496603075847098559510429, −3.77244562972407106360213876243, −2.66804047836263194591353122262, −2.26176015624340639066544856453, −0.61750858275570578847911970614,
1.15571204934586437826151436294, 1.74606248370178456779252796391, 3.09199679851013379680095574931, 4.19045128694439060141462000973, 5.11751060941928978195390488187, 6.02994354463711447345906345660, 6.42391657737435983604459445735, 7.71364139382464663298806198455, 8.338584600625961424429941979486, 8.856887158414794940244571787048