L(s) = 1 | + (0.809 + 2.48i)2-s + (−3.92 + 2.85i)4-s + (0.190 − 0.587i)5-s + (0.809 − 0.587i)7-s + (−6.04 − 4.39i)8-s + 1.61·10-s + (3.30 − 0.224i)11-s + (0.0729 + 0.224i)13-s + (2.11 + 1.53i)14-s + (3.04 − 9.37i)16-s + (0.354 − 1.08i)17-s + (−4.73 − 3.44i)19-s + (0.927 + 2.85i)20-s + (3.23 + 8.05i)22-s − 0.236·23-s + ⋯ |
L(s) = 1 | + (0.572 + 1.76i)2-s + (−1.96 + 1.42i)4-s + (0.0854 − 0.262i)5-s + (0.305 − 0.222i)7-s + (−2.13 − 1.55i)8-s + 0.511·10-s + (0.997 − 0.0676i)11-s + (0.0202 + 0.0622i)13-s + (0.566 + 0.411i)14-s + (0.761 − 2.34i)16-s + (0.0858 − 0.264i)17-s + (−1.08 − 0.789i)19-s + (0.207 + 0.637i)20-s + (0.689 + 1.71i)22-s − 0.0492·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.541549 + 1.09237i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.541549 + 1.09237i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (-3.30 + 0.224i)T \) |
good | 2 | \( 1 + (-0.809 - 2.48i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (-0.190 + 0.587i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-0.809 + 0.587i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-0.0729 - 0.224i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.354 + 1.08i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (4.73 + 3.44i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 0.236T + 23T^{2} \) |
| 29 | \( 1 + (4.85 - 3.52i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.88 + 5.79i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-5.04 + 3.66i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-0.190 - 0.138i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 + (8.16 + 5.93i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-0.118 - 0.363i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.97 + 4.33i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (3.57 - 10.9i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 - 1.85T + 67T^{2} \) |
| 71 | \( 1 + (3.19 - 9.82i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-4.61 + 3.35i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.39 - 10.4i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (0.454 - 1.40i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 8.23T + 89T^{2} \) |
| 97 | \( 1 + (-2.42 - 7.46i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−14.62401658294101409941727026059, −13.46890748198337438140272564225, −12.74354840792689086502648108786, −11.30104352816801899672395321546, −9.374689771408453280143777306937, −8.528606043985546045880646587795, −7.30266765817157788565595770418, −6.36893665835094932040845400721, −5.09379503607462220272508286657, −3.95878818678210987395774707797,
1.83112140241498648073084958194, 3.48286501579617997147708652733, 4.71167265570412505244281419165, 6.25377628042621661693132216485, 8.465909557683005444294389198268, 9.621233081045285182019871440769, 10.58914308488668776124766034680, 11.49579213424119554016335252283, 12.34212606160480480369532205912, 13.21903382862828714841697728728