| L(s) = 1 | + (−2.40 + 0.924i)2-s + (−1.01 − 1.40i)3-s + (3.45 − 3.11i)4-s + (−2.04 + 0.893i)5-s + (3.74 + 2.43i)6-s + (−0.860 + 3.21i)7-s + (−3.10 + 6.10i)8-s + (−0.928 + 2.85i)9-s + (4.11 − 4.04i)10-s + (1.47 − 3.30i)11-s + (−7.88 − 1.67i)12-s + (1.75 − 4.57i)13-s + (−0.896 − 8.52i)14-s + (3.33 + 1.96i)15-s + (0.873 − 8.31i)16-s + (1.89 + 0.966i)17-s + ⋯ |
| L(s) = 1 | + (−1.70 + 0.653i)2-s + (−0.587 − 0.809i)3-s + (1.72 − 1.55i)4-s + (−0.916 + 0.399i)5-s + (1.52 + 0.993i)6-s + (−0.325 + 1.21i)7-s + (−1.09 + 2.15i)8-s + (−0.309 + 0.950i)9-s + (1.30 − 1.27i)10-s + (0.444 − 0.997i)11-s + (−2.27 − 0.484i)12-s + (0.486 − 1.26i)13-s + (−0.239 − 2.27i)14-s + (0.861 + 0.507i)15-s + (0.218 − 2.07i)16-s + (0.460 + 0.234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.349939 - 0.0608707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.349939 - 0.0608707i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.01 + 1.40i)T \) |
| 5 | \( 1 + (2.04 - 0.893i)T \) |
| good | 2 | \( 1 + (2.40 - 0.924i)T + (1.48 - 1.33i)T^{2} \) |
| 7 | \( 1 + (0.860 - 3.21i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-1.47 + 3.30i)T + (-7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-1.75 + 4.57i)T + (-9.66 - 8.69i)T^{2} \) |
| 17 | \( 1 + (-1.89 - 0.966i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (0.281 - 0.0914i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.62 + 3.24i)T + (-4.78 - 22.4i)T^{2} \) |
| 29 | \( 1 + (-8.33 + 1.77i)T + (26.4 - 11.7i)T^{2} \) |
| 31 | \( 1 + (4.60 + 0.978i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (0.614 + 3.87i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-2.13 - 4.79i)T + (-27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-4.65 - 1.24i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-4.19 + 6.46i)T + (-19.1 - 42.9i)T^{2} \) |
| 53 | \( 1 + (-0.527 + 0.268i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (4.59 - 2.04i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (-9.16 - 4.08i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (1.61 + 2.48i)T + (-27.2 + 61.2i)T^{2} \) |
| 71 | \( 1 + (2.43 + 0.791i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.54 - 9.73i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (2.53 + 11.9i)T + (-72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (6.88 - 0.360i)T + (82.5 - 8.67i)T^{2} \) |
| 89 | \( 1 + (1.64 - 1.19i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-5.88 - 3.82i)T + (39.4 + 88.6i)T^{2} \) |
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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
−11.85478836352620406850056729949, −11.04993762086937034851319515161, −10.32200487829551721571585603249, −8.754471036929648342792991834111, −8.320385923177558375345829153639, −7.37157194771993364837785173203, −6.29832609594549149348060976636, −5.68357276634179363195195433351, −2.78715321653470391237141919558, −0.71891910637428540818333476999,
1.07243524193038034614507584994, 3.52680481337029901499774081206, 4.47818692794990733824513552666, 6.78011800684602344400441234077, 7.41457083602047743063839641138, 8.780278834989867596992655675249, 9.464278480516149436848705660159, 10.32605267612127445772005736366, 11.10618818399438018864005899267, 11.81845811439741570684398717627
