| L(s) = 1 | + 3-s − 4·5-s − 2·9-s + 3·11-s + 2·13-s − 4·15-s + 2·17-s + 6·23-s + 11·25-s − 5·27-s − 4·29-s − 10·31-s + 3·33-s + 2·37-s + 2·39-s + 9·41-s − 4·43-s + 8·45-s − 12·47-s − 7·49-s + 2·51-s − 2·53-s − 12·55-s − 59-s − 8·61-s − 8·65-s + 9·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 2/3·9-s + 0.904·11-s + 0.554·13-s − 1.03·15-s + 0.485·17-s + 1.25·23-s + 11/5·25-s − 0.962·27-s − 0.742·29-s − 1.79·31-s + 0.522·33-s + 0.328·37-s + 0.320·39-s + 1.40·41-s − 0.609·43-s + 1.19·45-s − 1.75·47-s − 49-s + 0.280·51-s − 0.274·53-s − 1.61·55-s − 0.130·59-s − 1.02·61-s − 0.992·65-s + 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−8.409991383292991495639050823044, −7.68070131627282443614947057547, −7.16868023884110097032904117960, −6.20437917308385561950056887329, −5.17163287819537530247393944939, −4.18215169002688948421186868912, −3.52024080459026738642689239885, −3.00588911083304856660566519275, −1.41714166109164809810127361557, 0,
1.41714166109164809810127361557, 3.00588911083304856660566519275, 3.52024080459026738642689239885, 4.18215169002688948421186868912, 5.17163287819537530247393944939, 6.20437917308385561950056887329, 7.16868023884110097032904117960, 7.68070131627282443614947057547, 8.409991383292991495639050823044
