L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.358 + 2.62i)7-s + 0.999·8-s + (−2.59 + 1.5i)11-s − 2.44·13-s + (−2.44 − i)14-s + (−0.5 + 0.866i)16-s + (5.12 − 2.95i)17-s + (5.12 + 2.95i)19-s − 3i·22-s + (2.12 − 3.67i)23-s + (1.22 − 2.12i)26-s + (2.09 − 1.62i)28-s + 7.24i·29-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.135 + 0.990i)7-s + 0.353·8-s + (−0.783 + 0.452i)11-s − 0.679·13-s + (−0.654 − 0.267i)14-s + (−0.125 + 0.216i)16-s + (1.24 − 0.717i)17-s + (1.17 + 0.678i)19-s − 0.639i·22-s + (0.442 − 0.766i)23-s + (0.240 − 0.416i)26-s + (0.395 − 0.306i)28-s + 1.34i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9971181338\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9971181338\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.358 - 2.62i)T \) |
good | 11 | \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.44T + 13T^{2} \) |
| 17 | \( 1 + (-5.12 + 2.95i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.12 - 2.95i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.12 + 3.67i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.24iT - 29T^{2} \) |
| 31 | \( 1 + (7.86 - 4.54i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.210 - 0.121i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 0.242iT - 43T^{2} \) |
| 47 | \( 1 + (5.12 + 2.95i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.62 - 6.27i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.03 + 6.98i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.878 - 0.507i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.66 - 5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.75iT - 71T^{2} \) |
| 73 | \( 1 + (-0.717 - 1.24i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (1.37 - 2.38i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6.63iT - 83T^{2} \) |
| 89 | \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−9.061191014325594727237089161496, −8.155326749511602659256105027257, −7.49582240599904201731707524539, −7.02446640263090839683550234728, −5.81113433686617261009865356083, −5.32411883628217648335473301347, −4.77913407573857216977377409521, −3.33179311236744859505928227049, −2.51364843617247239667898348552, −1.28141558610562689170863959346,
0.37980302078076454295286044726, 1.42381403937196745603907718074, 2.67182802459757513713878044661, 3.45914942624014748654530272389, 4.27836680682684524749952551898, 5.24596379436587874517004506483, 5.94042373043440272690822559502, 7.34600889313453639597071753036, 7.55185508136405821109247193417, 8.234833127675441162789077524117