| L(s) = 1 | + (−0.744 − 1.02i)2-s + (1.47 + 0.155i)3-s + (0.122 − 0.376i)4-s + (−0.940 − 1.62i)6-s + (−0.455 − 2.14i)7-s + (−2.88 + 0.937i)8-s + (−0.779 − 0.165i)9-s + (−0.636 − 0.706i)11-s + (0.238 − 0.536i)12-s + (−0.0683 − 0.153i)13-s + (−1.85 + 2.06i)14-s + (2.46 + 1.79i)16-s + (−4.88 − 4.40i)17-s + (0.410 + 0.921i)18-s + (1.05 + 0.468i)19-s + ⋯ |
| L(s) = 1 | + (−0.526 − 0.724i)2-s + (0.852 + 0.0895i)3-s + (0.0611 − 0.188i)4-s + (−0.383 − 0.664i)6-s + (−0.172 − 0.809i)7-s + (−1.02 + 0.331i)8-s + (−0.259 − 0.0552i)9-s + (−0.191 − 0.212i)11-s + (0.0689 − 0.154i)12-s + (−0.0189 − 0.0426i)13-s + (−0.495 + 0.550i)14-s + (0.617 + 0.448i)16-s + (−1.18 − 1.06i)17-s + (0.0967 + 0.217i)18-s + (0.241 + 0.107i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0215i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0100393 - 0.932361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0100393 - 0.932361i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + (-5.56 - 0.217i)T \) |
| good | 2 | \( 1 + (0.744 + 1.02i)T + (-0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-1.47 - 0.155i)T + (2.93 + 0.623i)T^{2} \) |
| 7 | \( 1 + (0.455 + 2.14i)T + (-6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (0.636 + 0.706i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.0683 + 0.153i)T + (-8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (4.88 + 4.40i)T + (1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.05 - 0.468i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (4.40 - 1.43i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.08 + 0.785i)T + (8.96 - 27.5i)T^{2} \) |
| 37 | \( 1 + (3.35 - 1.93i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0343 + 0.326i)T + (-40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (-3.91 + 8.79i)T + (-28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (-3.31 + 4.56i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.52 + 7.17i)T + (-48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (0.277 - 2.63i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + 1.74T + 61T^{2} \) |
| 67 | \( 1 + (0.478 + 0.276i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.11 + 0.236i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (5.88 - 5.30i)T + (7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (3.03 - 3.37i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.324 - 0.0341i)T + (81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-4.54 + 13.9i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-14.7 - 4.79i)T + (78.4 + 57.0i)T^{2} \) |
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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
−9.975529203067872269593918708685, −9.073145817873677063736379057704, −8.536354815389812149059344282335, −7.48457452478499464894427960177, −6.48846375989479086210156487255, −5.38396163301972372513085420266, −4.02934463641899443840285699298, −2.98472388144619464482513819873, −2.10991951376445546519512800760, −0.46217664115931073397362666920,
2.23120391756732440193552557331, 3.04536731970098543201655240763, 4.29276696733009105806631499087, 5.81889819949448136362809876215, 6.43812574812512257813328676010, 7.56990337690779497175871260783, 8.224825590874325193463456022816, 8.854942137674083864416260963983, 9.389688682726829102110983917007, 10.53849311767264931697490645255
