L(s) = 1 | − i·2-s − 4-s + (1.17 − 2.03i)5-s + i·8-s + (−2.03 − 1.17i)10-s + (−4.91 + 2.83i)11-s + (−1.48 + 0.859i)13-s + 16-s + (0.884 − 1.53i)17-s + (0.986 − 0.569i)19-s + (−1.17 + 2.03i)20-s + (2.83 + 4.91i)22-s + (−3.18 − 1.83i)23-s + (−0.259 − 0.449i)25-s + (0.859 + 1.48i)26-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.525 − 0.909i)5-s + 0.353i·8-s + (−0.643 − 0.371i)10-s + (−1.48 + 0.855i)11-s + (−0.413 + 0.238i)13-s + 0.250·16-s + (0.214 − 0.371i)17-s + (0.226 − 0.130i)19-s + (−0.262 + 0.454i)20-s + (0.605 + 1.04i)22-s + (−0.663 − 0.383i)23-s + (−0.0519 − 0.0899i)25-s + (0.168 + 0.292i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.616 - 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7682222611\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7682222611\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1.17 + 2.03i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (4.91 - 2.83i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.48 - 0.859i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.884 + 1.53i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.986 + 0.569i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.18 + 1.83i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.59 + 2.07i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 8.37iT - 31T^{2} \) |
| 37 | \( 1 + (-4.59 - 7.96i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.99 - 6.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.76 + 3.04i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 2.22T + 59T^{2} \) |
| 61 | \( 1 - 8.99iT - 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 + 4.52iT - 71T^{2} \) |
| 73 | \( 1 + (-4.62 - 2.67i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + (6.27 - 10.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.580 - 1.00i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.97 - 2.29i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.155132885771961757866228397777, −8.267541140319113304763697407816, −7.69258076761568429276926534163, −6.68328068672648009677614815576, −5.50985155696581638872282854768, −4.98621216825841189791138525467, −4.38632131739740998852070975436, −3.04335528143081711446795818548, −2.24401978863041721156712440618, −1.24638483084104275286309219579,
0.25148600788589473068100422893, 2.14207037400758165280400940063, 2.99813378782484880718481851608, 3.93775696372081723033644939895, 5.15741689416671929267030587158, 5.79685197325394871524676652198, 6.28038730932657388292589324959, 7.42193869568648613371691987406, 7.76005343971351958582020948730, 8.552989358559084303937250376164