L(s) = 1 | + (−1.73 − 0.0216i)3-s + (2.64 − 0.0973i)7-s + (2.99 + 0.0749i)9-s + (2.62 + 1.51i)11-s + 2.31i·13-s + (2.59 − 4.49i)17-s + (5.58 − 3.22i)19-s + (−4.58 + 0.111i)21-s + (−4.21 + 2.43i)23-s + (−5.19 − 0.194i)27-s + 3.48i·29-s + (1.16 + 0.673i)31-s + (−4.52 − 2.68i)33-s + (1.40 + 2.43i)37-s + (0.0501 − 4.01i)39-s + ⋯ |
L(s) = 1 | + (−0.999 − 0.0124i)3-s + (0.999 − 0.0368i)7-s + (0.999 + 0.0249i)9-s + (0.792 + 0.457i)11-s + 0.642i·13-s + (0.629 − 1.09i)17-s + (1.28 − 0.739i)19-s + (−0.999 + 0.0243i)21-s + (−0.879 + 0.507i)23-s + (−0.999 − 0.0374i)27-s + 0.647i·29-s + (0.209 + 0.120i)31-s + (−0.786 − 0.467i)33-s + (0.231 + 0.400i)37-s + (0.00803 − 0.642i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.587668129\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.587668129\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.73 + 0.0216i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.64 + 0.0973i)T \) |
good | 11 | \( 1 + (-2.62 - 1.51i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2.31iT - 13T^{2} \) |
| 17 | \( 1 + (-2.59 + 4.49i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.58 + 3.22i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.21 - 2.43i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.48iT - 29T^{2} \) |
| 31 | \( 1 + (-1.16 - 0.673i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.40 - 2.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 1.06T + 41T^{2} \) |
| 43 | \( 1 + 2.42T + 43T^{2} \) |
| 47 | \( 1 + (1.90 + 3.30i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.11 + 3.52i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.18 - 10.7i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-11.4 + 6.60i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.14 - 3.72i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.08iT - 71T^{2} \) |
| 73 | \( 1 + (0.132 + 0.0763i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.04 - 5.27i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + (-7.10 - 12.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8.63iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.410878172248298855817394282429, −8.251597125214503740261499496671, −7.30449001232220780393227128340, −6.93639864091092866080028913107, −5.86430547985532501506076437140, −5.02617572762926501898004969794, −4.56403671531440046137479025140, −3.44494483103533946552477893518, −1.89918643788382804246493809808, −0.979436566753245479013592819685,
0.909360930457597723955542637645, 1.82537548755106696259447529768, 3.47923203732569809063258158053, 4.27255360822106937244416474778, 5.23511149221286141968029889349, 5.88129018160128197689449557093, 6.50048973058921374516075806482, 7.81469155178523592959054831241, 7.949246215970590889850237460212, 9.153749835318282961272404695392