L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s − 3.18·5-s + 0.999·8-s + (1.59 + 2.75i)10-s − 3.18·11-s + (−2.85 − 4.93i)13-s + (−0.5 − 0.866i)16-s + (−0.760 − 1.31i)17-s + (0.641 − 1.11i)19-s + (1.59 − 2.75i)20-s + (1.59 + 2.75i)22-s − 2.23·23-s + 5.12·25-s + (−2.85 + 4.93i)26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s − 1.42·5-s + 0.353·8-s + (0.503 + 0.871i)10-s − 0.959·11-s + (−0.790 − 1.36i)13-s + (−0.125 − 0.216i)16-s + (−0.184 − 0.319i)17-s + (0.147 − 0.254i)19-s + (0.355 − 0.616i)20-s + (0.339 + 0.587i)22-s − 0.466·23-s + 1.02·25-s + (−0.559 + 0.968i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.831 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3389612216\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3389612216\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 3.18T + 5T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 + (2.85 + 4.93i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.760 + 1.31i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.641 + 1.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 + (-3.54 + 6.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.71 - 8.15i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.80 + 4.85i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.41 + 5.91i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.02 + 1.78i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.562 + 0.974i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.56 - 2.70i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.48 - 9.49i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 8.69T + 71T^{2} \) |
| 73 | \( 1 + (-2.48 - 4.30i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.03 - 6.98i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.112 + 0.195i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.42 - 12.8i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.779127240378728089597193349798, −8.169253615967829571973361462078, −7.56876014660706277319458041535, −7.08139662374109657281127776112, −5.61929337299538755113011175085, −4.87679592062663476605062096929, −4.00127049199699868644782249842, −3.12820886430964922037372299198, −2.42045039607319416795710394791, −0.68223168755965770565846557042,
0.20147753699830051876794276380, 1.88913144548316399530390403789, 3.17120428188905476849618366368, 4.26062710464139396347368206101, 4.71383326266261165764143884865, 5.78254439363188828428069777656, 6.73037940901259963244379778293, 7.45474842852149164922660666420, 7.86453506399553109595055245051, 8.623994988741571189472076544247