| L(s) = 1 | − i·2-s − 4-s − i·7-s + i·8-s + i·9-s + (−1 + i)11-s − 14-s + 16-s + 18-s + (1 + i)22-s − i·25-s + i·28-s + (−1 − i)29-s − i·32-s − i·36-s + (−1 + i)37-s + ⋯ |
| L(s) = 1 | − i·2-s − 4-s − i·7-s + i·8-s + i·9-s + (−1 + i)11-s − 14-s + 16-s + 18-s + (1 + i)22-s − i·25-s + i·28-s + (−1 − i)29-s − i·32-s − i·36-s + (−1 + i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5314225250\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5314225250\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - iT^{2} \) |
| 5 | \( 1 + iT^{2} \) |
| 11 | \( 1 + (1 - i)T - iT^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - iT^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (1 + i)T + iT^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1 - i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-1 + i)T - iT^{2} \) |
| 59 | \( 1 + iT^{2} \) |
| 61 | \( 1 - iT^{2} \) |
| 67 | \( 1 + (-1 - i)T + iT^{2} \) |
| 71 | \( 1 - 2iT - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−13.45286471755867347856734524811, −12.75795018895155456044755509988, −11.49301810311495345341512613038, −10.42149304749246711383091061266, −9.968337038803622567918569441583, −8.319056652516069500791959709910, −7.34148884483076194341677592755, −5.22631067709461204170447629166, −4.13114150743211819775629176162, −2.27803974715281569194064287612,
3.39688143770569238837513800590, 5.28799620715987171801146708145, 6.09862088320331059470531847771, 7.48250950853313495465445459749, 8.712784745011879189339577700650, 9.372527297341524736564459728269, 10.90122155092304599712070645171, 12.31230408122088277357194352773, 13.13005281311943114418937395804, 14.30069270320800848660558005175
