L(s) = 1 | + (0.674 − 1.24i)2-s + (−0.839 − 3.13i)3-s + (−1.08 − 1.67i)4-s + (0.734 − 2.74i)5-s + (−4.46 − 1.07i)6-s + (−2.81 + 0.223i)8-s + (−6.51 + 3.76i)9-s + (−2.91 − 2.76i)10-s + (4.32 − 1.15i)11-s + (−4.33 + 4.82i)12-s + (0.558 − 0.558i)13-s − 9.20·15-s + (−1.62 + 3.65i)16-s + (1.09 − 1.89i)17-s + (0.279 + 10.6i)18-s + (2.55 + 0.684i)19-s + ⋯ |
L(s) = 1 | + (0.477 − 0.878i)2-s + (−0.484 − 1.80i)3-s + (−0.544 − 0.838i)4-s + (0.328 − 1.22i)5-s + (−1.82 − 0.436i)6-s + (−0.996 + 0.0788i)8-s + (−2.17 + 1.25i)9-s + (−0.920 − 0.873i)10-s + (1.30 − 0.349i)11-s + (−1.25 + 1.39i)12-s + (0.154 − 0.154i)13-s − 2.37·15-s + (−0.406 + 0.913i)16-s + (0.266 − 0.460i)17-s + (0.0659 + 2.50i)18-s + (0.585 + 0.156i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.910049 + 1.31907i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.910049 + 1.31907i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.674 + 1.24i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.839 + 3.13i)T + (-2.59 + 1.5i)T^{2} \) |
| 5 | \( 1 + (-0.734 + 2.74i)T + (-4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-4.32 + 1.15i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.558 + 0.558i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.09 + 1.89i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.55 - 0.684i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (-0.647 + 0.374i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.07 - 3.07i)T - 29iT^{2} \) |
| 31 | \( 1 + (-4.43 + 7.67i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.69 + 6.33i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 0.267iT - 41T^{2} \) |
| 43 | \( 1 + (-4.53 - 4.53i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.74 - 6.48i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.36 + 1.16i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (7.07 - 1.89i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-5.73 - 1.53i)T + (52.8 + 30.5i)T^{2} \) |
| 67 | \( 1 + (-1.48 - 5.53i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 9.37iT - 71T^{2} \) |
| 73 | \( 1 + (9.08 + 5.24i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.10 - 8.84i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (0.474 - 0.474i)T - 83iT^{2} \) |
| 89 | \( 1 + (12.5 - 7.26i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−9.580145254649306546797000863595, −8.937495730885811747960504804388, −8.044372635048673991097816569211, −6.92763646690980663254546000325, −5.89039470122428488517346374395, −5.48573784940422153017677170132, −4.20067981262560015144526616755, −2.65583067265013610596682473963, −1.39682824672883887001440914591, −0.845071586432049567917674672165,
3.02719755430473890659219888947, 3.77032975531153694992014006662, 4.58049923647379682571992739662, 5.60638731174130028685510945317, 6.34589814020301264750639478133, 7.06088550939033360653776745530, 8.496331988209638565848692127396, 9.323092112154266209324582093272, 9.953454425083414982354031577947, 10.74997582791420255648506511091