L(s) = 1 | − 1.41i·5-s + 2.82i·7-s − 4·11-s − 2·13-s − 1.41i·17-s + 5.65i·19-s + 4·23-s + 2.99·25-s − 7.07i·29-s − 8.48i·31-s + 4.00·35-s − 8·37-s − 4.24i·41-s − 11.3i·43-s − 12·47-s + ⋯ |
L(s) = 1 | − 0.632i·5-s + 1.06i·7-s − 1.20·11-s − 0.554·13-s − 0.342i·17-s + 1.29i·19-s + 0.834·23-s + 0.599·25-s − 1.31i·29-s − 1.52i·31-s + 0.676·35-s − 1.31·37-s − 0.662i·41-s − 1.72i·43-s − 1.75·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6795022032\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6795022032\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 1.41iT - 5T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 7.07iT - 29T^{2} \) |
| 31 | \( 1 + 8.48iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 + 4.24iT - 41T^{2} \) |
| 43 | \( 1 + 11.3iT - 43T^{2} \) |
| 47 | \( 1 + 12T + 47T^{2} \) |
| 53 | \( 1 + 12.7iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 5.65iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 8T + 73T^{2} \) |
| 79 | \( 1 + 2.82iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 15.5iT - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.650370595617985789180098837586, −8.104566962140610598435152614983, −7.34150211670380816578073557738, −6.26777613181024333962562965269, −5.30477956450098129495367544897, −5.11265373585282500258274198123, −3.84043259140866042327596939201, −2.70281061195281806164723263700, −1.92829210324510103586957248605, −0.23054144447577228874751391699,
1.31870632846192158168050730455, 2.83941373417171295977862401120, 3.26131021009050861886019039244, 4.75394705023948498917572289366, 5.00009400216933859675251187869, 6.40355310819957039614955280145, 7.06911132119200124551994124987, 7.50861622863333305290099105873, 8.469740669072292414134061951165, 9.283076350652407326201495881785