L(s) = 1 | + (−2.41 + 1.39i)2-s + 1.64·3-s + (2.89 − 5.01i)4-s + (1.92 − 1.11i)5-s + (−3.97 + 2.29i)6-s + (3.37 − 1.95i)7-s + 10.6i·8-s − 0.293·9-s + (−3.11 + 5.38i)10-s + (0.708 + 0.409i)11-s + (4.76 − 8.25i)12-s + (−3.50 + 0.848i)13-s + (−5.44 + 9.43i)14-s + (3.17 − 1.83i)15-s + (−9.00 − 15.5i)16-s + (1.44 + 2.50i)17-s + ⋯ |
L(s) = 1 | + (−1.70 + 0.987i)2-s + 0.949·3-s + (1.44 − 2.50i)4-s + (0.863 − 0.498i)5-s + (−1.62 + 0.937i)6-s + (1.27 − 0.737i)7-s + 3.74i·8-s − 0.0977·9-s + (−0.983 + 1.70i)10-s + (0.213 + 0.123i)11-s + (1.37 − 2.38i)12-s + (−0.971 + 0.235i)13-s + (−1.45 + 2.52i)14-s + (0.819 − 0.473i)15-s + (−2.25 − 3.89i)16-s + (0.350 + 0.607i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.06323 + 0.136022i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.06323 + 0.136022i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (3.50 - 0.848i)T \) |
| 31 | \( 1 + (2.32 + 5.05i)T \) |
good | 2 | \( 1 + (2.41 - 1.39i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 - 1.64T + 3T^{2} \) |
| 5 | \( 1 + (-1.92 + 1.11i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-3.37 + 1.95i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.708 - 0.409i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.44 - 2.50i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.61 + 3.81i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.34 + 2.33i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.996 + 1.72i)T + (-14.5 + 25.1i)T^{2} \) |
| 37 | \( 1 - 7.35iT - 37T^{2} \) |
| 41 | \( 1 + (-2.60 - 1.50i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.65 + 2.86i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 6.33iT - 47T^{2} \) |
| 53 | \( 1 + (0.0147 + 0.0255i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.804 + 0.464i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.79 - 4.84i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.24 - 2.45i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.10iT - 71T^{2} \) |
| 73 | \( 1 + (-12.9 + 7.46i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.61 - 6.26i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.90 - 2.25i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (11.2 - 6.47i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.6 + 6.12i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.91650843700507470244620270294, −9.718079569143924546171438773339, −9.488316130311320170456808904001, −8.448106114643009787409677093246, −7.82223365651301771353104768266, −7.13612998814092034341044101592, −5.78090376979113444299659756413, −4.86564153407738332364622606554, −2.31763192704553406640868381015, −1.27022532526652782155327437515,
1.69046247453408311896307081746, 2.47566250316986253747865629832, 3.40306207434790718985105711788, 5.53164255141738694639167199439, 7.25444989781456281382828571542, 7.87887076698568103853094344133, 8.683056650974810774793499253201, 9.493373907605797397653673496205, 9.975163638460386915752299442855, 11.09275820770227786764259711588