| L(s) = 1 | − 7-s + 6·11-s − 2·13-s − 4·19-s − 6·23-s − 5·25-s + 6·29-s − 8·31-s − 2·37-s − 12·41-s − 4·43-s + 12·47-s + 49-s − 6·53-s + 10·61-s + 8·67-s + 6·71-s − 10·73-s − 6·77-s + 4·79-s + 12·83-s − 12·89-s + 2·91-s − 10·97-s − 12·101-s − 8·103-s + 6·107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.80·11-s − 0.554·13-s − 0.917·19-s − 1.25·23-s − 25-s + 1.11·29-s − 1.43·31-s − 0.328·37-s − 1.87·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s + 1.28·61-s + 0.977·67-s + 0.712·71-s − 1.17·73-s − 0.683·77-s + 0.450·79-s + 1.31·83-s − 1.27·89-s + 0.209·91-s − 1.01·97-s − 1.19·101-s − 0.788·103-s + 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.213446609484277420706681145366, −7.18052525439954131151018372494, −6.61214442273728283164556360378, −6.01676531608062216159532650200, −5.07643921282730531606229802841, −3.99784334743039531176458960720, −3.70585619782695729246427373887, −2.36003185284708036230417844213, −1.49990868331367069680950997074, 0,
1.49990868331367069680950997074, 2.36003185284708036230417844213, 3.70585619782695729246427373887, 3.99784334743039531176458960720, 5.07643921282730531606229802841, 6.01676531608062216159532650200, 6.61214442273728283164556360378, 7.18052525439954131151018372494, 8.213446609484277420706681145366
