L(s) = 1 | + (1.02 + 0.970i)2-s + (1.62 + 0.609i)3-s + (0.116 + 1.99i)4-s + (0.5 − 0.866i)5-s + (1.07 + 2.20i)6-s + (−0.518 + 0.299i)7-s + (−1.81 + 2.16i)8-s + (2.25 + 1.97i)9-s + (1.35 − 0.405i)10-s + (0.700 − 0.404i)11-s + (−1.02 + 3.30i)12-s + (−3.52 − 2.03i)13-s + (−0.824 − 0.195i)14-s + (1.33 − 1.09i)15-s + (−3.97 + 0.466i)16-s − 1.87i·17-s + ⋯ |
L(s) = 1 | + (0.727 + 0.686i)2-s + (0.936 + 0.351i)3-s + (0.0583 + 0.998i)4-s + (0.223 − 0.387i)5-s + (0.439 + 0.898i)6-s + (−0.196 + 0.113i)7-s + (−0.642 + 0.766i)8-s + (0.752 + 0.658i)9-s + (0.428 − 0.128i)10-s + (0.211 − 0.121i)11-s + (−0.296 + 0.955i)12-s + (−0.977 − 0.564i)13-s + (−0.220 − 0.0522i)14-s + (0.345 − 0.283i)15-s + (−0.993 + 0.116i)16-s − 0.453i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.96287 + 1.61298i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.96287 + 1.61298i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.02 - 0.970i)T \) |
| 3 | \( 1 + (-1.62 - 0.609i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
good | 7 | \( 1 + (0.518 - 0.299i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.700 + 0.404i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.52 + 2.03i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.87iT - 17T^{2} \) |
| 19 | \( 1 - 3.32T + 19T^{2} \) |
| 23 | \( 1 + (-1.00 + 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.38 - 2.40i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.04 + 2.33i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 7.30iT - 37T^{2} \) |
| 41 | \( 1 + (7.32 + 4.22i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.19 + 7.26i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.76 - 11.7i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 5.27T + 53T^{2} \) |
| 59 | \( 1 + (-6.41 - 3.70i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.38 - 2.53i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.68 + 11.5i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.24T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 + (-3.85 + 2.22i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.04 + 2.33i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 9.41iT - 89T^{2} \) |
| 97 | \( 1 + (-1.95 - 3.38i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.03481577077845787868490051244, −10.64646877027978157919019677978, −9.470105621764929095511945037366, −8.837547444396866783978483782865, −7.75110692533170027781226775861, −7.05817213916878691590810860116, −5.58854293486330730808102558559, −4.75488487003556542188661171748, −3.55888642907503505342492528755, −2.47085233479394382519844532900,
1.68064793375904463957475818866, 2.85828985598237419026572961460, 3.84447400635747428126778816534, 5.10482865037178080927635210908, 6.53355499374850869327326266180, 7.22132637923308443435703748842, 8.614115649849949999889968047873, 9.751868792091606015755695463838, 10.08679397632838923895186177217, 11.52314121406140222270423634139