L(s) = 1 | + (1.36 + 1.36i)5-s + (0.866 + 0.5i)13-s + (−1.5 + 0.866i)17-s + 2.73i·25-s + (−0.5 + 0.866i)29-s + (−0.5 − 1.86i)37-s + (0.5 − 0.133i)41-s + (−0.866 − 0.5i)49-s + 1.73·53-s + (−0.5 − 0.866i)61-s + (0.499 + 1.86i)65-s + (1.36 − 1.36i)73-s + (−3.23 − 0.866i)85-s + (0.366 + 1.36i)89-s + (−0.366 + 1.36i)97-s + ⋯ |
L(s) = 1 | + (1.36 + 1.36i)5-s + (0.866 + 0.5i)13-s + (−1.5 + 0.866i)17-s + 2.73i·25-s + (−0.5 + 0.866i)29-s + (−0.5 − 1.86i)37-s + (0.5 − 0.133i)41-s + (−0.866 − 0.5i)49-s + 1.73·53-s + (−0.5 − 0.866i)61-s + (0.499 + 1.86i)65-s + (1.36 − 1.36i)73-s + (−3.23 − 0.866i)85-s + (0.366 + 1.36i)89-s + (−0.366 + 1.36i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.606078303\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.606078303\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
| 7 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (0.5 + 1.86i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 - 1.73T + T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + (-1.36 + 1.36i)T - iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \) |
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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
−9.128295438345657898650823209140, −8.149736568755642208798595538406, −6.99284452007462247397879609377, −6.71256610975758741749800948139, −5.97060806732334382579285842214, −5.36478469346673152443473375799, −4.09462543380708907128234012449, −3.35029872704169760064589517363, −2.25098775176741486836304515018, −1.76803876258856167420237023971,
0.927465546532444714823775508251, 1.91654682853477902141441577748, 2.78750935751617561829731001680, 4.16079570648368290269094092284, 4.82380161556296108571273617122, 5.54426826023992215053489874283, 6.17749515906433761699346016013, 6.88383563423910810096999855236, 8.090171427533214929140697918990, 8.636816389299437736296342269869