L(s) = 1 | + (0.687 − 1.23i)2-s − 0.614i·3-s + (−1.05 − 1.69i)4-s + (−2.07 − 0.832i)5-s + (−0.759 − 0.422i)6-s + (2.83 + 2.83i)7-s + (−2.82 + 0.134i)8-s + 2.62·9-s + (−2.45 + 1.99i)10-s + (1.95 + 1.95i)11-s + (−1.04 + 0.647i)12-s − 2.05·13-s + (5.45 − 1.55i)14-s + (−0.511 + 1.27i)15-s + (−1.77 + 3.58i)16-s + (−4.06 − 4.06i)17-s + ⋯ |
L(s) = 1 | + (0.486 − 0.873i)2-s − 0.354i·3-s + (−0.527 − 0.849i)4-s + (−0.928 − 0.372i)5-s + (−0.310 − 0.172i)6-s + (1.07 + 1.07i)7-s + (−0.998 + 0.0473i)8-s + 0.874·9-s + (−0.776 + 0.630i)10-s + (0.590 + 0.590i)11-s + (−0.301 + 0.187i)12-s − 0.569·13-s + (1.45 − 0.415i)14-s + (−0.132 + 0.329i)15-s + (−0.444 + 0.895i)16-s + (−0.986 − 0.986i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.104 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.812951 - 0.732058i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.812951 - 0.732058i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.687 + 1.23i)T \) |
| 5 | \( 1 + (2.07 + 0.832i)T \) |
good | 3 | \( 1 + 0.614iT - 3T^{2} \) |
| 7 | \( 1 + (-2.83 - 2.83i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.95 - 1.95i)T + 11iT^{2} \) |
| 13 | \( 1 + 2.05T + 13T^{2} \) |
| 17 | \( 1 + (4.06 + 4.06i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.683 - 0.683i)T + 19iT^{2} \) |
| 23 | \( 1 + (4.95 - 4.95i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.835 - 0.835i)T - 29iT^{2} \) |
| 31 | \( 1 + 2.35iT - 31T^{2} \) |
| 37 | \( 1 + 4.54T + 37T^{2} \) |
| 41 | \( 1 + 5.07iT - 41T^{2} \) |
| 43 | \( 1 + 0.849T + 43T^{2} \) |
| 47 | \( 1 + (-2.72 + 2.72i)T - 47iT^{2} \) |
| 53 | \( 1 + 5.17iT - 53T^{2} \) |
| 59 | \( 1 + (-4.16 + 4.16i)T - 59iT^{2} \) |
| 61 | \( 1 + (-5.55 - 5.55i)T + 61iT^{2} \) |
| 67 | \( 1 + 1.73T + 67T^{2} \) |
| 71 | \( 1 - 2.33T + 71T^{2} \) |
| 73 | \( 1 + (-4.39 - 4.39i)T + 73iT^{2} \) |
| 79 | \( 1 + 14.0T + 79T^{2} \) |
| 83 | \( 1 - 2.75iT - 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + (3.52 + 3.52i)T + 97iT^{2} \) |
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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
−14.03258096595873744640418610776, −12.73242113511172795928592907099, −11.91373869665796178434521418583, −11.43281578419205298351996257676, −9.755211554658424139736326603860, −8.630440152697391424305050682451, −7.21104873352215385582307598686, −5.22958028383356760367181170569, −4.17535766716212241244810693878, −1.95813183540523251651850835743,
3.93047030287500441640760317790, 4.58162580949885439806168679542, 6.63123374325369996593243073490, 7.61125761896255316115618066976, 8.585104660251373589008184281129, 10.37718459681003917039633564326, 11.44137856047894984991471362114, 12.64279417238456249228974427633, 13.95361815109619134226275436106, 14.68741333576407925443563310353