L(s) = 1 | + (1.14 + 0.833i)2-s + 0.586i·3-s + (0.609 + 1.90i)4-s + 5-s + (−0.489 + 0.670i)6-s + (−2.52 + 0.792i)7-s + (−0.892 + 2.68i)8-s + 2.65·9-s + (1.14 + 0.833i)10-s − 0.580·11-s + (−1.11 + 0.357i)12-s + 1.14·13-s + (−3.54 − 1.20i)14-s + 0.586i·15-s + (−3.25 + 2.32i)16-s + 1.82i·17-s + ⋯ |
L(s) = 1 | + (0.807 + 0.589i)2-s + 0.338i·3-s + (0.304 + 0.952i)4-s + 0.447·5-s + (−0.199 + 0.273i)6-s + (−0.954 + 0.299i)7-s + (−0.315 + 0.948i)8-s + 0.885·9-s + (0.361 + 0.263i)10-s − 0.174·11-s + (−0.322 + 0.103i)12-s + 0.318·13-s + (−0.947 − 0.320i)14-s + 0.151i·15-s + (−0.814 + 0.580i)16-s + 0.443i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0171 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0171 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41287 + 1.38883i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41287 + 1.38883i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.14 - 0.833i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (2.52 - 0.792i)T \) |
good | 3 | \( 1 - 0.586iT - 3T^{2} \) |
| 11 | \( 1 + 0.580T + 11T^{2} \) |
| 13 | \( 1 - 1.14T + 13T^{2} \) |
| 17 | \( 1 - 1.82iT - 17T^{2} \) |
| 19 | \( 1 + 4.72iT - 19T^{2} \) |
| 23 | \( 1 + 2.79iT - 23T^{2} \) |
| 29 | \( 1 + 7.40iT - 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 + 4.35iT - 37T^{2} \) |
| 41 | \( 1 - 8.46iT - 41T^{2} \) |
| 43 | \( 1 + 4.32T + 43T^{2} \) |
| 47 | \( 1 - 3.56T + 47T^{2} \) |
| 53 | \( 1 + 5.48iT - 53T^{2} \) |
| 59 | \( 1 + 13.2iT - 59T^{2} \) |
| 61 | \( 1 - 0.275T + 61T^{2} \) |
| 67 | \( 1 + 7.71T + 67T^{2} \) |
| 71 | \( 1 - 1.13iT - 71T^{2} \) |
| 73 | \( 1 + 1.78iT - 73T^{2} \) |
| 79 | \( 1 - 10.0iT - 79T^{2} \) |
| 83 | \( 1 - 15.6iT - 83T^{2} \) |
| 89 | \( 1 + 15.3iT - 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−12.44180191135931529877047562576, −11.27973432849057124539933750982, −10.13493344652970640673219204864, −9.259802190413267247780403813356, −8.120750607575287187376966897564, −6.82956476661737994668117983688, −6.19004580816843283533885925915, −4.96405854128223254473993056555, −3.88759940770095881841811451882, −2.57211465962240082266948683814,
1.44054566451780328956572257013, 3.02988721233242942861211599516, 4.18211898017596451031509117885, 5.54062887264364700109207579635, 6.51746580042564101577294455540, 7.36699191188161189052797189083, 9.089595305881780830138672450245, 10.09104783379427561721449497603, 10.54030326679972479304088891251, 11.96439006625236204447312032570