L(s) = 1 | + (−3.80 + 3.80i)2-s − 12.9i·4-s + (−6.56 − 24.1i)5-s + (16.6 − 16.6i)7-s + (−11.6 − 11.6i)8-s + (116. + 66.7i)10-s + 215.·11-s + (−29.9 − 29.9i)13-s + 126. i·14-s + 295.·16-s + (3.97 − 3.97i)17-s − 604. i·19-s + (−311. + 84.9i)20-s + (−818. + 818. i)22-s + (−376. − 376. i)23-s + ⋯ |
L(s) = 1 | + (−0.950 + 0.950i)2-s − 0.808i·4-s + (−0.262 − 0.964i)5-s + (0.339 − 0.339i)7-s + (−0.182 − 0.182i)8-s + (1.16 + 0.667i)10-s + 1.77·11-s + (−0.177 − 0.177i)13-s + 0.644i·14-s + 1.15·16-s + (0.0137 − 0.0137i)17-s − 1.67i·19-s + (−0.779 + 0.212i)20-s + (−1.69 + 1.69i)22-s + (−0.711 − 0.711i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.958 + 0.283i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.831734 - 0.120471i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.831734 - 0.120471i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (6.56 + 24.1i)T \) |
good | 2 | \( 1 + (3.80 - 3.80i)T - 16iT^{2} \) |
| 7 | \( 1 + (-16.6 + 16.6i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 - 215.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (29.9 + 29.9i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-3.97 + 3.97i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + 604. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (376. + 376. i)T + 2.79e5iT^{2} \) |
| 29 | \( 1 + 624. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 263.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-979. + 979. i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.57e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (30.8 + 30.8i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (1.78e3 - 1.78e3i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (707. + 707. i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 - 1.36e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.98e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (986. - 986. i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 + 68.0T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-2.47e3 - 2.47e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 - 6.04e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (-5.15e3 - 5.15e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 + 7.02e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.08e4 + 1.08e4i)T - 8.85e7iT^{2} \) |
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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
−15.38017118794153135043631292736, −14.19612791544485433007779403774, −12.64636843105268172224473904748, −11.44141749130610746891047575294, −9.556828985538710610858438676394, −8.788220463060621843873178006167, −7.60696475657327907921158999599, −6.29602641274894401920615070815, −4.34401598142625114555860784365, −0.798289300451765372052626030605,
1.71143705428645798906169085396, 3.58548807063401939161266522531, 6.23241761615639168187215979850, 7.937400100767493557076628656245, 9.288549895813705477385506835756, 10.30485465520781565862380753565, 11.55023607343392023280095318713, 12.05705809649070947308145302902, 14.24794480217943376055480510384, 14.87664250505277829600095515292