L(s) = 1 | − 2·2-s + 2·4-s + (−2 − i)5-s − 3·7-s + (4 + 2i)10-s + 5i·11-s + (3 + 2i)13-s + 6·14-s − 4·16-s + 3i·17-s − 6i·19-s + (−4 − 2i)20-s − 10i·22-s − 9i·23-s + (3 + 4i)25-s + (−6 − 4i)26-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s + (−0.894 − 0.447i)5-s − 1.13·7-s + (1.26 + 0.632i)10-s + 1.50i·11-s + (0.832 + 0.554i)13-s + 1.60·14-s − 16-s + 0.727i·17-s − 1.37i·19-s + (−0.894 − 0.447i)20-s − 2.13i·22-s − 1.87i·23-s + (0.600 + 0.800i)25-s + (−1.17 − 0.784i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.336697 - 0.195393i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.336697 - 0.195393i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (2 + i)T \) |
| 13 | \( 1 + (-3 - 2i)T \) |
good | 2 | \( 1 + 2T + 2T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 5iT - 11T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 9iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 3T + 37T^{2} \) |
| 41 | \( 1 + 5iT - 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 5iT - 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 15T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - iT - 89T^{2} \) |
| 97 | \( 1 + 3T + 97T^{2} \) |
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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
−10.42103320353477661117268803697, −9.476220323931995388585989725842, −8.883656010439182856886203348173, −8.162329390411607941758449411778, −7.00690631551313728286812764995, −6.66019463738317538948276633695, −4.78970329042366362349592707627, −3.85500934471812827628489769915, −2.17463805380279945586825658032, −0.47700115344610429901397330196,
0.947733204992314286482106005486, 3.06116115712673195544296332274, 3.77184443352862720855340195537, 5.70234336970779568459039934288, 6.59748566259394238960804034940, 7.65728275570204550665991032186, 8.182446764583122457814635297813, 9.111382214782530781019327809046, 9.863840180864464806878309789231, 10.77354353536073679881650218281