| L(s) = 1 | − 557. i·3-s + 1.55e4·5-s + 8.13e4i·7-s + 2.21e5·9-s − 1.83e6i·11-s − 1.54e6·13-s − 8.66e6i·15-s + 4.03e7·17-s − 8.21e7i·19-s + 4.53e7·21-s − 1.41e8i·23-s − 2.19e6·25-s − 4.19e8i·27-s + 4.11e8·29-s + 1.49e9i·31-s + ⋯ |
| L(s) = 1 | − 0.764i·3-s + 0.995·5-s + 0.691i·7-s + 0.415·9-s − 1.03i·11-s − 0.320·13-s − 0.760i·15-s + 1.66·17-s − 1.74i·19-s + 0.528·21-s − 0.956i·23-s − 0.00900·25-s − 1.08i·27-s + 0.692·29-s + 1.67i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.94545 - 1.12320i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94545 - 1.12320i\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 557. iT - 5.31e5T^{2} \) |
| 5 | \( 1 - 1.55e4T + 2.44e8T^{2} \) |
| 7 | \( 1 - 8.13e4iT - 1.38e10T^{2} \) |
| 11 | \( 1 + 1.83e6iT - 3.13e12T^{2} \) |
| 13 | \( 1 + 1.54e6T + 2.32e13T^{2} \) |
| 17 | \( 1 - 4.03e7T + 5.82e14T^{2} \) |
| 19 | \( 1 + 8.21e7iT - 2.21e15T^{2} \) |
| 23 | \( 1 + 1.41e8iT - 2.19e16T^{2} \) |
| 29 | \( 1 - 4.11e8T + 3.53e17T^{2} \) |
| 31 | \( 1 - 1.49e9iT - 7.87e17T^{2} \) |
| 37 | \( 1 + 2.50e9T + 6.58e18T^{2} \) |
| 41 | \( 1 + 1.24e9T + 2.25e19T^{2} \) |
| 43 | \( 1 - 6.93e9iT - 3.99e19T^{2} \) |
| 47 | \( 1 + 8.07e9iT - 1.16e20T^{2} \) |
| 53 | \( 1 - 1.93e10T + 4.91e20T^{2} \) |
| 59 | \( 1 - 8.78e9iT - 1.77e21T^{2} \) |
| 61 | \( 1 - 8.42e10T + 2.65e21T^{2} \) |
| 67 | \( 1 - 2.92e10iT - 8.18e21T^{2} \) |
| 71 | \( 1 - 2.13e11iT - 1.64e22T^{2} \) |
| 73 | \( 1 + 2.24e11T + 2.29e22T^{2} \) |
| 79 | \( 1 + 2.95e10iT - 5.90e22T^{2} \) |
| 83 | \( 1 + 5.72e10iT - 1.06e23T^{2} \) |
| 89 | \( 1 + 4.16e11T + 2.46e23T^{2} \) |
| 97 | \( 1 + 7.23e11T + 6.93e23T^{2} \) |
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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
−16.02127767255819079982479203141, −14.30152890612116698756816417620, −13.21644847071322941134654794989, −12.02017048909731559806536491868, −10.17041986001780058252371394441, −8.626543547951990850556857418043, −6.80865073081310027745236009201, −5.41221012899246348955409120268, −2.65735151191530007205372038082, −1.05692748200235443445236408270,
1.60384788240128593125623589121, 3.88852964264781699188446764584, 5.53002041080969843686740229407, 7.48738185326094804221182461037, 9.841674046956035964091665810768, 10.12285783093264346118012774615, 12.31242871165040092766396487418, 13.82782440356921175560209350387, 14.99557258197934202982582908089, 16.50524083248688291677390550899
