| L(s) = 1 | + (−3.52 − 3.52i)7-s − 3i·9-s + 2.15·11-s + (−3.98 − 3.98i)17-s + 4.35i·19-s + (−6.35 + 6.35i)23-s + (1.71 − 1.71i)43-s + (9.67 + 9.67i)47-s + 17.8i·49-s − 15.1·61-s + (−10.5 + 10.5i)63-s + (−6.91 + 6.91i)73-s + (−7.58 − 7.58i)77-s − 9·81-s + (3.64 − 3.64i)83-s + ⋯ |
| L(s) = 1 | + (−1.33 − 1.33i)7-s − i·9-s + 0.648·11-s + (−0.966 − 0.966i)17-s + 0.999i·19-s + (−1.32 + 1.32i)23-s + (0.260 − 0.260i)43-s + (1.41 + 1.41i)47-s + 2.55i·49-s − 1.94·61-s + (−1.33 + 1.33i)63-s + (−0.809 + 0.809i)73-s + (−0.864 − 0.864i)77-s − 81-s + (0.399 − 0.399i)83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.850 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1403745739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1403745739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 + 3iT^{2} \) |
| 7 | \( 1 + (3.52 + 3.52i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 + 13iT^{2} \) |
| 17 | \( 1 + (3.98 + 3.98i)T + 17iT^{2} \) |
| 23 | \( 1 + (6.35 - 6.35i)T - 23iT^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (-1.71 + 1.71i)T - 43iT^{2} \) |
| 47 | \( 1 + (-9.67 - 9.67i)T + 47iT^{2} \) |
| 53 | \( 1 + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15.1T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (6.91 - 6.91i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + (-3.64 + 3.64i)T - 83iT^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97iT^{2} \) |
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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
−9.093911103476372681793935115366, −7.75375201019735803494514140408, −7.12645553746385852805578293479, −6.40240787865130434011268884772, −5.82782653275933265016066770184, −4.24643365436069808438980424807, −3.83694250863501078512697545091, −2.94860099252022039977998306974, −1.30237332840607077818305559133, −0.05140036112512253824700489735,
2.07349772811157219445186345195, 2.67977852577209151747034676895, 3.89278848341773106933061346828, 4.80479772784458978027949878456, 5.92500084557265997622301769323, 6.32675406267855021963754666303, 7.19803502543848581018563638507, 8.416696536883918107071468723359, 8.805123338703873063049396259518, 9.570915659408174438854017818548
