L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−2.16 − 0.575i)5-s + 0.999i·6-s + (−1.29 + 0.746i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−1.57 + 1.58i)10-s − 4.99·11-s + (0.866 + 0.499i)12-s + (3.55 + 6.16i)13-s + 1.49i·14-s + (2.15 − 0.581i)15-s + (−0.5 + 0.866i)16-s + (2.24 − 3.88i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.966 − 0.257i)5-s + 0.408i·6-s + (−0.488 + 0.282i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.499 + 0.500i)10-s − 1.50·11-s + (0.249 + 0.144i)12-s + (0.987 + 1.70i)13-s + 0.399i·14-s + (0.557 − 0.150i)15-s + (−0.125 + 0.216i)16-s + (0.544 − 0.942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.101007613\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.101007613\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (2.16 + 0.575i)T \) |
| 37 | \( 1 + (-4.87 + 3.64i)T \) |
good | 7 | \( 1 + (1.29 - 0.746i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 4.99T + 11T^{2} \) |
| 13 | \( 1 + (-3.55 - 6.16i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.24 + 3.88i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.68 + 1.55i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 7.83T + 23T^{2} \) |
| 29 | \( 1 + 8.24iT - 29T^{2} \) |
| 31 | \( 1 - 0.925iT - 31T^{2} \) |
| 41 | \( 1 + (-3.23 - 5.60i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 8.04T + 43T^{2} \) |
| 47 | \( 1 - 4.51iT - 47T^{2} \) |
| 53 | \( 1 + (-3.66 - 2.11i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.55 + 2.63i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-8.38 + 4.83i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.5 + 6.65i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.259 - 0.448i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 13.8iT - 73T^{2} \) |
| 79 | \( 1 + (0.836 - 0.482i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.83 - 4.52i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.79 - 3.92i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 9.48T + 97T^{2} \) |
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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
−9.723241256935066560688298774819, −9.264839478183697763598471263967, −8.215601665015733302615436533705, −7.23925936029805279847650773367, −6.30371910610784745134814633699, −5.18852825493596378130987497628, −4.60448392729224923669240574535, −3.56792987595577529613187797314, −2.63999476128362572113422567577, −0.821562796179361951458184484692,
0.72507937397459379640176675874, 3.07787579634467752835353336043, 3.56165425786780214427405877254, 5.04936391123911043139455589454, 5.55020404172263386516383048256, 6.59355476499264768846784315501, 7.44599867005794767398596245667, 7.992553530728477402536128617635, 8.675243908656265485468867498077, 10.31237904143602741638304223763