| L(s) = 1 | − 2·4-s − 2·7-s − 3·11-s + 4·13-s + 4·16-s + 6·17-s − 19-s + 6·23-s + 4·28-s − 9·29-s − 31-s − 8·37-s + 3·41-s + 4·43-s + 6·44-s − 12·47-s − 3·49-s − 8·52-s − 6·53-s + 3·59-s − 10·61-s − 8·64-s − 14·67-s − 12·68-s − 3·71-s − 2·73-s + 2·76-s + ⋯ |
| L(s) = 1 | − 4-s − 0.755·7-s − 0.904·11-s + 1.10·13-s + 16-s + 1.45·17-s − 0.229·19-s + 1.25·23-s + 0.755·28-s − 1.67·29-s − 0.179·31-s − 1.31·37-s + 0.468·41-s + 0.609·43-s + 0.904·44-s − 1.75·47-s − 3/7·49-s − 1.10·52-s − 0.824·53-s + 0.390·59-s − 1.28·61-s − 64-s − 1.71·67-s − 1.45·68-s − 0.356·71-s − 0.234·73-s + 0.229·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.911588909676083735140008198739, −7.996251614910726055453610263734, −7.38323020306752051711450229099, −6.18776174727003732201040896108, −5.53159145215903046087701254220, −4.77166689679712781581501695027, −3.56661227356653752031129287676, −3.15988881675676159750407403400, −1.40254687148271888788317255475, 0,
1.40254687148271888788317255475, 3.15988881675676159750407403400, 3.56661227356653752031129287676, 4.77166689679712781581501695027, 5.53159145215903046087701254220, 6.18776174727003732201040896108, 7.38323020306752051711450229099, 7.996251614910726055453610263734, 8.911588909676083735140008198739
