| L(s) = 1 | + (−0.471 − 1.75i)2-s + (0.0524 − 0.195i)3-s + (0.592 − 0.341i)4-s + (−1.13 − 4.87i)5-s − 0.369·6-s + (−0.563 + 6.97i)7-s + (−6.03 − 6.03i)8-s + (7.75 + 4.47i)9-s + (−8.03 + 4.28i)10-s + (6.67 + 11.5i)11-s + (−0.0358 − 0.133i)12-s + (4.91 + 4.91i)13-s + (12.5 − 2.29i)14-s + (−1.01 − 0.0338i)15-s + (−6.39 + 11.0i)16-s + (−19.4 − 5.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.235 − 0.879i)2-s + (0.0174 − 0.0652i)3-s + (0.148 − 0.0854i)4-s + (−0.226 − 0.974i)5-s − 0.0615·6-s + (−0.0804 + 0.996i)7-s + (−0.753 − 0.753i)8-s + (0.862 + 0.497i)9-s + (−0.803 + 0.428i)10-s + (0.606 + 1.05i)11-s + (−0.00298 − 0.0111i)12-s + (0.377 + 0.377i)13-s + (0.895 − 0.164i)14-s + (−0.0675 − 0.00225i)15-s + (−0.399 + 0.692i)16-s + (−1.14 − 0.306i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.756305 - 0.630301i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.756305 - 0.630301i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.13 + 4.87i)T \) |
| 7 | \( 1 + (0.563 - 6.97i)T \) |
| good | 2 | \( 1 + (0.471 + 1.75i)T + (-3.46 + 2i)T^{2} \) |
| 3 | \( 1 + (-0.0524 + 0.195i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (-6.67 - 11.5i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-4.91 - 4.91i)T + 169iT^{2} \) |
| 17 | \( 1 + (19.4 + 5.20i)T + (250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (14.0 + 8.11i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-18.7 + 5.02i)T + (458. - 264.5i)T^{2} \) |
| 29 | \( 1 + 29.5iT - 841T^{2} \) |
| 31 | \( 1 + (-9.49 - 16.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-1.92 - 7.17i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 52.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + (11.3 + 11.3i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-14.2 - 53.3i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (3.27 - 12.2i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-12.0 + 6.92i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-57.3 + 99.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-56.0 - 15.0i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 + 86.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-6.50 + 24.2i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (23.0 + 13.3i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-32.6 - 32.6i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (62.6 + 36.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-47.8 + 47.8i)T - 9.40e3iT^{2} \) |
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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
−15.86074378962991218611255837879, −15.24704115054663704021879877960, −13.13140069949016601261941445186, −12.31461428789155865450236964652, −11.30080174447260769775616891548, −9.739514883712820143025610041285, −8.766454054337996167590179056655, −6.69833054982626240803528046313, −4.56885454584650333410437033323, −1.94018280982684620414559032909,
3.64834189542641045314735576157, 6.38799088761916484036878754360, 7.13148881847001597954376525747, 8.602825393517548231010113061561, 10.45121635405183419811833210123, 11.46630716738592321635527554306, 13.26467723263697313754500703161, 14.60249204188828962355072583592, 15.45574188975510236753319414414, 16.51920401999327307725274266878
