L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.448 + 1.67i)3-s + (0.866 − 0.499i)4-s + 1.73i·6-s + (−0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (−1.73 − 1.00i)9-s + (0.448 + 1.67i)12-s + i·14-s + (0.500 − 0.866i)16-s + (−1.93 − 0.517i)18-s + (−1.50 − 0.866i)21-s + (−0.258 − 0.965i)23-s + (0.866 + 1.5i)24-s + (1.22 − 1.22i)27-s + (0.258 + 0.965i)28-s + ⋯ |
L(s) = 1 | + (0.965 − 0.258i)2-s + (−0.448 + 1.67i)3-s + (0.866 − 0.499i)4-s + 1.73i·6-s + (−0.258 + 0.965i)7-s + (0.707 − 0.707i)8-s + (−1.73 − 1.00i)9-s + (0.448 + 1.67i)12-s + i·14-s + (0.500 − 0.866i)16-s + (−1.93 − 0.517i)18-s + (−1.50 − 0.866i)21-s + (−0.258 − 0.965i)23-s + (0.866 + 1.5i)24-s + (1.22 − 1.22i)27-s + (0.258 + 0.965i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.394685283\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.394685283\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.258 - 0.965i)T \) |
good | 3 | \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \) |
| 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 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 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 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 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
−10.72402301863372233970943256509, −10.28607976040705329089981407997, −9.337821263478711338493481284866, −8.589581196910798735308856067003, −6.93947372539664195270563660466, −5.83718016116654840148847588693, −5.34109147541218438055301842866, −4.40510366695514855028385869107, −3.54966038497482328875417548074, −2.49328113025985959878365685198,
1.41964317179010972080787827297, 2.73088878189420697766663729175, 4.02795197713944862910319567984, 5.29148411484387431366615563296, 6.25559622452298971139573877461, 6.78192031468000646887469572965, 7.67590624654212101181122828828, 8.083113296062513697235440927042, 9.765077021344419031182412793342, 11.02957297752899223526044502221