L(s) = 1 | + 0.792·3-s + i·5-s + (−0.792 + 2.52i)7-s − 2.37·9-s + 0.792i·11-s + 5.37i·13-s + 0.792i·15-s − 3.37i·17-s + 3.46·19-s + (−0.627 + 2i)21-s − 1.87i·23-s − 25-s − 4.25·27-s + 5.37·29-s − 8.51·31-s + ⋯ |
L(s) = 1 | + 0.457·3-s + 0.447i·5-s + (−0.299 + 0.954i)7-s − 0.790·9-s + 0.238i·11-s + 1.49i·13-s + 0.204i·15-s − 0.817i·17-s + 0.794·19-s + (−0.136 + 0.436i)21-s − 0.391i·23-s − 0.200·25-s − 0.819·27-s + 0.997·29-s − 1.52·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9241154039\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9241154039\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (0.792 - 2.52i)T \) |
good | 3 | \( 1 - 0.792T + 3T^{2} \) |
| 11 | \( 1 - 0.792iT - 11T^{2} \) |
| 13 | \( 1 - 5.37iT - 13T^{2} \) |
| 17 | \( 1 + 3.37iT - 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 + 1.87iT - 23T^{2} \) |
| 29 | \( 1 - 5.37T + 29T^{2} \) |
| 31 | \( 1 + 8.51T + 31T^{2} \) |
| 37 | \( 1 - 0.744T + 37T^{2} \) |
| 41 | \( 1 + 2.74iT - 41T^{2} \) |
| 43 | \( 1 - 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + 6.63T + 59T^{2} \) |
| 61 | \( 1 - 0.744iT - 61T^{2} \) |
| 67 | \( 1 - 6.63iT - 67T^{2} \) |
| 71 | \( 1 + 6.63iT - 71T^{2} \) |
| 73 | \( 1 + 2.74iT - 73T^{2} \) |
| 79 | \( 1 - 14.0iT - 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 17.4iT - 89T^{2} \) |
| 97 | \( 1 - 2.11iT - 97T^{2} \) |
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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
−9.378176910970264200767337609165, −8.731318321033163417965895055019, −7.927030017724001672771358577304, −6.99444541656982700494895027335, −6.35730226838477357418813328199, −5.46936533657421645619196143308, −4.60890365596856585522608413418, −3.39723564419530370096073152584, −2.72439388245439149759730316225, −1.83047091297745045650366766625,
0.28394487742746485842902385082, 1.55687443235877722866599115349, 3.12097510560604680922931731132, 3.44373831069238384260059347159, 4.66352965057292916225887749518, 5.55917740040895117090758296745, 6.23594412877000822938054349039, 7.38994900037063348336550377348, 7.976790226022694639906757624429, 8.553061272697279386793479902799