L(s) = 1 | + (−1.31 + 0.761i)2-s + (−1.49 + 2.59i)3-s + (0.160 − 0.277i)4-s + (2.55 − 1.47i)5-s − 4.56i·6-s − 2.55i·8-s + (−2.98 − 5.17i)9-s + (−2.24 + 3.88i)10-s + (2.05 + 1.18i)11-s + (0.480 + 0.831i)12-s + (2.32 + 2.75i)13-s + 8.82i·15-s + (2.26 + 3.93i)16-s + (2.68 − 4.65i)17-s + (7.87 + 4.54i)18-s + (4.63 − 2.67i)19-s + ⋯ |
L(s) = 1 | + (−0.932 + 0.538i)2-s + (−0.864 + 1.49i)3-s + (0.0801 − 0.138i)4-s + (1.14 − 0.658i)5-s − 1.86i·6-s − 0.904i·8-s + (−0.995 − 1.72i)9-s + (−0.709 + 1.22i)10-s + (0.619 + 0.357i)11-s + (0.138 + 0.240i)12-s + (0.645 + 0.763i)13-s + 2.27i·15-s + (0.567 + 0.982i)16-s + (0.651 − 1.12i)17-s + (1.85 + 1.07i)18-s + (1.06 − 0.614i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.525524 + 0.677057i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.525524 + 0.677057i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-2.32 - 2.75i)T \) |
good | 2 | \( 1 + (1.31 - 0.761i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1.49 - 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.55 + 1.47i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.05 - 1.18i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.68 + 4.65i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.63 + 2.67i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.39 - 2.41i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 0.585T + 29T^{2} \) |
| 31 | \( 1 + (-7.21 - 4.16i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.585 + 0.337i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.2iT - 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + (-0.465 + 0.268i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.95 - 6.84i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.32 - 3.07i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.39 - 2.41i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.64 - 2.10i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.25iT - 71T^{2} \) |
| 73 | \( 1 + (9.40 + 5.43i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.14 + 5.44i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 1.74iT - 83T^{2} \) |
| 89 | \( 1 + (-8.86 + 5.11i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 9.90iT - 97T^{2} \) |
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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
−10.32832098936478179457460438264, −9.788107813170862274417298368115, −9.190763780663685489310236596792, −8.799425785413722046108349696565, −7.21717665391946573696213821899, −6.29827914264466570746697009513, −5.35363811081244411861859002455, −4.61248054949407649887745330824, −3.43631428340104486348906946737, −1.07966564497352313426924796944,
1.01382900580084368348306935751, 1.74105136365210773698716683521, 2.99259239195127363649354375155, 5.33723547986845053254775782914, 6.06759400391498384811420571942, 6.55879882192035849625703228465, 7.901735248279287857905348926539, 8.404594872663841222485855427444, 9.842894709890034605920075622420, 10.23290374829550344187694818314