| L(s) = 1 | + 1.73i·2-s − 0.999·4-s + 2i·7-s + 1.73i·8-s − 3.46·11-s + i·13-s − 3.46·14-s − 5·16-s + 5.19i·17-s − 2·19-s − 5.99i·22-s − 3.46i·23-s − 1.73·26-s − 1.99i·28-s − 1.73·29-s + ⋯ |
| L(s) = 1 | + 1.22i·2-s − 0.499·4-s + 0.755i·7-s + 0.612i·8-s − 1.04·11-s + 0.277i·13-s − 0.925·14-s − 1.25·16-s + 1.26i·17-s − 0.458·19-s − 1.27i·22-s − 0.722i·23-s − 0.339·26-s − 0.377i·28-s − 0.321·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8367451332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8367451332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 1.73iT - 2T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - iT - 13T^{2} \) |
| 17 | \( 1 - 5.19iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 7iT - 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 2iT - 43T^{2} \) |
| 47 | \( 1 - 6.92iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 7T + 61T^{2} \) |
| 67 | \( 1 + 10iT - 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 7iT - 73T^{2} \) |
| 79 | \( 1 + 2T + 79T^{2} \) |
| 83 | \( 1 - 13.8iT - 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.349199224898288688071826117064, −8.553260256015047482922953279984, −8.115631938457230520559388568738, −7.35251477860364532749020610531, −6.36254490457928315677313940755, −5.95256969491012050268713111413, −5.09816717814197307883664807277, −4.29385597405379900920398936898, −2.85065386380197381335095136149, −1.96365714280001234364633736269,
0.28079742699792462762690322188, 1.48321088283183700866022534230, 2.69384006054574773533303821647, 3.26635679337664497482225881798, 4.38024644182308467261085656773, 5.07325631206401795385858994105, 6.28363569961192941036428684525, 7.18584362619360875855465935977, 7.80848376464542406599212336638, 8.832455530923911914586473091155
