| L(s) = 1 | + 2·3-s + 4·7-s + 9-s + 2·11-s − 17-s + 8·21-s + 4·23-s − 5·25-s − 4·27-s + 4·29-s − 4·31-s + 4·33-s − 12·37-s − 2·41-s − 4·43-s + 9·49-s − 2·51-s + 2·53-s − 4·59-s + 4·63-s − 4·67-s + 8·69-s + 2·73-s − 10·75-s + 8·77-s + 4·79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.242·17-s + 1.74·21-s + 0.834·23-s − 25-s − 0.769·27-s + 0.742·29-s − 0.718·31-s + 0.696·33-s − 1.97·37-s − 0.312·41-s − 0.609·43-s + 9/7·49-s − 0.280·51-s + 0.274·53-s − 0.520·59-s + 0.503·63-s − 0.488·67-s + 0.963·69-s + 0.234·73-s − 1.15·75-s + 0.911·77-s + 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−13.99664331485679, −13.83978063957576, −13.35263924145923, −12.63901728553120, −12.02583028098517, −11.67951774326727, −11.13148328867640, −10.69554814796468, −10.07581771337510, −9.456805253462319, −8.936157182332828, −8.577138947090490, −8.192636939941833, −7.655501588347117, −7.150407630995685, −6.631680713875070, −5.803788700107541, −5.226115916393839, −4.762557685114925, −4.069072216165055, −3.598499204362134, −2.973899719940063, −2.230518666795577, −1.723327109669255, −1.242302871953915, 0,
1.242302871953915, 1.723327109669255, 2.230518666795577, 2.973899719940063, 3.598499204362134, 4.069072216165055, 4.762557685114925, 5.226115916393839, 5.803788700107541, 6.631680713875070, 7.150407630995685, 7.655501588347117, 8.192636939941833, 8.577138947090490, 8.936157182332828, 9.456805253462319, 10.07581771337510, 10.69554814796468, 11.13148328867640, 11.67951774326727, 12.02583028098517, 12.63901728553120, 13.35263924145923, 13.83978063957576, 13.99664331485679
