L(s) = 1 | + (−1.12 + 1.31i)3-s − 4.34·7-s + (−0.448 − 2.96i)9-s + 1.83i·11-s + 0.588·13-s − 5.37·17-s − 5.38·19-s + (4.90 − 5.70i)21-s + 2.40i·23-s + (4.40 + 2.76i)27-s + 7.98·29-s − 7.06i·31-s + (−2.41 − 2.07i)33-s − 2.72·37-s + (−0.664 + 0.772i)39-s + ⋯ |
L(s) = 1 | + (−0.652 + 0.758i)3-s − 1.64·7-s + (−0.149 − 0.988i)9-s + 0.553i·11-s + 0.163·13-s − 1.30·17-s − 1.23·19-s + (1.07 − 1.24i)21-s + 0.502i·23-s + (0.847 + 0.531i)27-s + 1.48·29-s − 1.26i·31-s + (−0.419 − 0.361i)33-s − 0.447·37-s + (−0.106 + 0.123i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6419243526\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6419243526\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.12 - 1.31i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 4.34T + 7T^{2} \) |
| 11 | \( 1 - 1.83iT - 11T^{2} \) |
| 13 | \( 1 - 0.588T + 13T^{2} \) |
| 17 | \( 1 + 5.37T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 - 2.40iT - 23T^{2} \) |
| 29 | \( 1 - 7.98T + 29T^{2} \) |
| 31 | \( 1 + 7.06iT - 31T^{2} \) |
| 37 | \( 1 + 2.72T + 37T^{2} \) |
| 41 | \( 1 - 3.42iT - 41T^{2} \) |
| 43 | \( 1 - 2.96iT - 43T^{2} \) |
| 47 | \( 1 + 9.81iT - 47T^{2} \) |
| 53 | \( 1 - 6.65iT - 53T^{2} \) |
| 59 | \( 1 - 10.7iT - 59T^{2} \) |
| 61 | \( 1 + 9.27iT - 61T^{2} \) |
| 67 | \( 1 + 4.13iT - 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 + 4.42iT - 73T^{2} \) |
| 79 | \( 1 + 12.5iT - 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 4.21iT - 89T^{2} \) |
| 97 | \( 1 - 2.16iT - 97T^{2} \) |
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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
−9.194425844372090890114590304779, −8.418148867581075058802645724288, −7.11765969668105900461241162478, −6.39235444683495133835233330407, −6.10315154544686583743274245701, −4.86414465999959883997912598016, −4.17297004584732087947811278199, −3.36341450578847632820317805683, −2.30131883439537066980726749948, −0.38912863445346678337883892681,
0.64966780003792171014766914761, 2.18828027723895500910986331323, 3.05664233232493554954153958645, 4.16769049407143587210585282472, 5.15994812893576096537001596862, 6.25715874639803859508646082132, 6.50942130712889246572673447773, 7.07667113031970192787206798498, 8.381202920729653772129904962219, 8.745945513427000517160554114229