L(s) = 1 | + 0.601i·5-s + 0.163i·11-s + 3.37i·13-s + 2.13i·17-s − 0.953·19-s + 6.99i·23-s + 4.63·25-s + 1.28·29-s − 3.35·31-s + 5.16·37-s − 6.67i·41-s + 6.09i·43-s − 3.69·47-s − 13.8·53-s − 0.0984·55-s + ⋯ |
L(s) = 1 | + 0.269i·5-s + 0.0493i·11-s + 0.937i·13-s + 0.517i·17-s − 0.218·19-s + 1.45i·23-s + 0.927·25-s + 0.239·29-s − 0.602·31-s + 0.848·37-s − 1.04i·41-s + 0.928i·43-s − 0.539·47-s − 1.90·53-s − 0.0132·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.145212762\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145212762\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 0.601iT - 5T^{2} \) |
| 11 | \( 1 - 0.163iT - 11T^{2} \) |
| 13 | \( 1 - 3.37iT - 13T^{2} \) |
| 17 | \( 1 - 2.13iT - 17T^{2} \) |
| 19 | \( 1 + 0.953T + 19T^{2} \) |
| 23 | \( 1 - 6.99iT - 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 + 3.35T + 31T^{2} \) |
| 37 | \( 1 - 5.16T + 37T^{2} \) |
| 41 | \( 1 + 6.67iT - 41T^{2} \) |
| 43 | \( 1 - 6.09iT - 43T^{2} \) |
| 47 | \( 1 + 3.69T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 - 8.05T + 59T^{2} \) |
| 61 | \( 1 - 0.181iT - 61T^{2} \) |
| 67 | \( 1 + 9.09iT - 67T^{2} \) |
| 71 | \( 1 - 3.36iT - 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 6.98iT - 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 14.8iT - 89T^{2} \) |
| 97 | \( 1 - 9.08iT - 97T^{2} \) |
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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
−8.098956214169790073913330857428, −7.54737388063498268553179007682, −6.74125482318808092718760950066, −6.26088919212630282017519508811, −5.38544205924423361408381225181, −4.64474213834503811753804083040, −3.84551456416079203923739751588, −3.12872694528468332750167011858, −2.11120936878162735630896366569, −1.29198549749055693292983273933,
0.28793838843528625985427694828, 1.28409871162856649955518525659, 2.53359358528390617640406527505, 3.09552481555009910503155657514, 4.16677230644514584511511056478, 4.82519993098486517692984289326, 5.50311802332402452509429691014, 6.29814456581087927585102611974, 6.95550532499607977595414757978, 7.70634991384526586020136967370