L(s) = 1 | + 2.64i·5-s + 8.30i·7-s + 18.7·11-s − 22.3i·13-s − 23.4·17-s + 9.93·19-s − 32.3i·23-s + 18.0·25-s − 8.45i·29-s − 46.9i·31-s − 21.9·35-s + 28.3i·37-s + 77.7·41-s − 58.4·43-s − 54.2i·47-s + ⋯ |
L(s) = 1 | + 0.528i·5-s + 1.18i·7-s + 1.70·11-s − 1.71i·13-s − 1.38·17-s + 0.522·19-s − 1.40i·23-s + 0.721·25-s − 0.291i·29-s − 1.51i·31-s − 0.626·35-s + 0.765i·37-s + 1.89·41-s − 1.35·43-s − 1.15i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.125279620\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.125279620\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2.64iT - 25T^{2} \) |
| 7 | \( 1 - 8.30iT - 49T^{2} \) |
| 11 | \( 1 - 18.7T + 121T^{2} \) |
| 13 | \( 1 + 22.3iT - 169T^{2} \) |
| 17 | \( 1 + 23.4T + 289T^{2} \) |
| 19 | \( 1 - 9.93T + 361T^{2} \) |
| 23 | \( 1 + 32.3iT - 529T^{2} \) |
| 29 | \( 1 + 8.45iT - 841T^{2} \) |
| 31 | \( 1 + 46.9iT - 961T^{2} \) |
| 37 | \( 1 - 28.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 77.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 58.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + 54.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 5.81iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 47.5T + 3.48e3T^{2} \) |
| 61 | \( 1 - 27.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 50.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 7.34iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 29.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 43.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 10.7T + 6.88e3T^{2} \) |
| 89 | \( 1 - 136.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 54.6T + 9.40e3T^{2} \) |
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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
−8.941453609167403474987472164735, −8.544332278060279974082372498060, −7.50828538916472968235766622410, −6.47702192245212936387239434860, −6.09931209799354204411112079702, −5.05169850801204488345005297148, −4.04318984745663763638244493591, −2.96257352017741886398999710278, −2.21967866117901880485918390575, −0.67829378021738169268568044390,
1.04776613325914870916574091514, 1.76511987725358158763964369159, 3.47748490947260825091165759329, 4.24395944823088973357835368308, 4.75121338267335511258665451285, 6.14077565411463728614944139579, 6.94715720236407793785288014470, 7.26407632320704690467590457437, 8.663554810891593956655413655401, 9.153513071167926557924391197901