L(s) = 1 | − 2.62·5-s − 3.45i·7-s − 3.79i·11-s − 0.312i·13-s − 4.13·17-s + (3.41 − 2.70i)19-s − 1.93i·23-s + 1.87·25-s − 5.99i·29-s + 9.38·31-s + 9.06i·35-s + 6.64i·37-s − 12.1i·41-s − 10.2i·43-s − 2.71i·47-s + ⋯ |
L(s) = 1 | − 1.17·5-s − 1.30i·7-s − 1.14i·11-s − 0.0866i·13-s − 1.00·17-s + (0.783 − 0.621i)19-s − 0.402i·23-s + 0.375·25-s − 1.11i·29-s + 1.68·31-s + 1.53i·35-s + 1.09i·37-s − 1.89i·41-s − 1.56i·43-s − 0.396i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8630152499\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8630152499\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3.41 + 2.70i)T \) |
good | 5 | \( 1 + 2.62T + 5T^{2} \) |
| 7 | \( 1 + 3.45iT - 7T^{2} \) |
| 11 | \( 1 + 3.79iT - 11T^{2} \) |
| 13 | \( 1 + 0.312iT - 13T^{2} \) |
| 17 | \( 1 + 4.13T + 17T^{2} \) |
| 23 | \( 1 + 1.93iT - 23T^{2} \) |
| 29 | \( 1 + 5.99iT - 29T^{2} \) |
| 31 | \( 1 - 9.38T + 31T^{2} \) |
| 37 | \( 1 - 6.64iT - 37T^{2} \) |
| 41 | \( 1 + 12.1iT - 41T^{2} \) |
| 43 | \( 1 + 10.2iT - 43T^{2} \) |
| 47 | \( 1 + 2.71iT - 47T^{2} \) |
| 53 | \( 1 - 7.29iT - 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.54T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 + 2.45T + 79T^{2} \) |
| 83 | \( 1 - 4.38iT - 83T^{2} \) |
| 89 | \( 1 + 5.30iT - 89T^{2} \) |
| 97 | \( 1 + 10.6iT - 97T^{2} \) |
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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
−7.79043599377970878079268622676, −7.11808623150333380940088701285, −6.59384438951266456115425982456, −5.62400416670075456064425028539, −4.60158823705557543917914158962, −4.07574911312015775022302192848, −3.43976775652658163566956237423, −2.51638683199143523793015715149, −0.908755238859582112277813071989, −0.29341278938123923230701285613,
1.41714767054459674449571320439, 2.47530499857341838123112491184, 3.21703675783773357548519061938, 4.24527589977896040184899440591, 4.77345047946934087635761086751, 5.60135695159803915050495916059, 6.49247860187054978236326386573, 7.12510475890559165300120091196, 8.015734515851712843580242055096, 8.278275415916506780395663716432