L(s) = 1 | + (0.366 − 0.366i)5-s + (0.866 − 0.5i)9-s + (0.866 + 0.5i)13-s + (−0.5 + 0.866i)17-s + 0.732i·25-s + (0.5 − 0.133i)29-s + (0.133 + 0.5i)37-s + (1.86 − 0.5i)41-s + (0.133 − 0.5i)45-s + (−0.866 − 0.5i)49-s − 1.73i·53-s + (−1.86 − 0.5i)61-s + (0.5 − 0.133i)65-s + (−1.36 + 1.36i)73-s + (0.499 − 0.866i)81-s + ⋯ |
L(s) = 1 | + (0.366 − 0.366i)5-s + (0.866 − 0.5i)9-s + (0.866 + 0.5i)13-s + (−0.5 + 0.866i)17-s + 0.732i·25-s + (0.5 − 0.133i)29-s + (0.133 + 0.5i)37-s + (1.86 − 0.5i)41-s + (0.133 − 0.5i)45-s + (−0.866 − 0.5i)49-s − 1.73i·53-s + (−1.86 − 0.5i)61-s + (0.5 − 0.133i)65-s + (−1.36 + 1.36i)73-s + (0.499 − 0.866i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.528029430\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.528029430\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + (-0.866 - 0.5i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
good | 3 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 5 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
| 7 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 37 | \( 1 + (-0.133 - 0.5i)T + (-0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + 1.73iT - T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 73 | \( 1 + (1.36 - 1.36i)T - iT^{2} \) |
| 79 | \( 1 + iT^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \) |
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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
−8.822171166282762684903189488897, −8.076877994442241782808800680682, −7.18127618158181998473112585353, −6.42708999408752574556777848930, −5.90032178215830543248602271594, −4.83082661343267252436901643703, −4.12262601156797750176801894205, −3.36685376128482842510757183104, −2.01200081749381880998069491998, −1.20437912669463358398065554136,
1.16639595903471030800521402821, 2.33060199183292897044637170764, 3.12722208512587809263406871312, 4.28743561787703092616917186461, 4.82429089575324629435273148887, 5.98598972745594129458514995769, 6.38148339963609185222930062352, 7.46799594824834858286185368702, 7.78280445697162819272437114051, 8.917732094905913128261490770043