L(s) = 1 | + (−0.258 + 0.965i)2-s + (−1.67 + 0.448i)3-s + (−0.866 − 0.499i)4-s − 1.73i·6-s + (0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (1.73 − 1.00i)9-s + (1.67 + 0.448i)12-s + i·14-s + (0.500 + 0.866i)16-s + (0.517 + 1.93i)18-s + (−1.50 + 0.866i)21-s + (0.965 + 0.258i)23-s + (−0.866 + 1.5i)24-s + (−1.22 + 1.22i)27-s + (−0.965 − 0.258i)28-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (−1.67 + 0.448i)3-s + (−0.866 − 0.499i)4-s − 1.73i·6-s + (0.965 − 0.258i)7-s + (0.707 − 0.707i)8-s + (1.73 − 1.00i)9-s + (1.67 + 0.448i)12-s + i·14-s + (0.500 + 0.866i)16-s + (0.517 + 1.93i)18-s + (−1.50 + 0.866i)21-s + (0.965 + 0.258i)23-s + (−0.866 + 1.5i)24-s + (−1.22 + 1.22i)27-s + (−0.965 − 0.258i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5068991207\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5068991207\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.965 + 0.258i)T \) |
good | 3 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 - iT - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 - 1.73iT - T^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 89 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + iT^{2} \) |
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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
−10.87083808084454894222439647146, −10.08842716723380580370944650744, −9.192796193895111243066348901148, −8.111045211667389999536515263412, −7.12314797637691466104237241172, −6.43086522906765847450344312310, −5.34330453163173520981801563796, −4.97034235313518680131673916278, −3.98174293552171557589671434373, −1.16108403800718222692595285252,
1.02094521967963520497415484235, 2.29020810053866576943474903554, 4.13176739420658555138731924910, 5.03111826911873762098200910908, 5.71164254098171852564892061842, 6.97066544893672630325941258823, 7.85635099086691726635904007936, 8.853250077216533587747650773405, 9.964011272102307446663476139760, 10.83322722981388516316262553952