| L(s) = 1 | − 5-s − 7-s + 4·11-s − 2·13-s − 2·17-s + 4·19-s − 8·23-s + 25-s − 6·29-s + 8·31-s + 35-s − 2·37-s − 2·41-s + 12·43-s − 8·47-s + 49-s − 6·53-s − 4·55-s + 4·59-s − 2·61-s + 2·65-s − 12·67-s + 8·71-s − 14·73-s − 4·77-s + 12·83-s + 2·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.169·35-s − 0.328·37-s − 0.312·41-s + 1.82·43-s − 1.16·47-s + 1/7·49-s − 0.824·53-s − 0.539·55-s + 0.520·59-s − 0.256·61-s + 0.248·65-s − 1.46·67-s + 0.949·71-s − 1.63·73-s − 0.455·77-s + 1.31·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.80736164148569186936146869381, −7.21269514981927104076803165456, −6.41219830495361519372956586712, −5.86187005182510651321447425475, −4.79858245819537333446038366323, −4.08876702135675126351599181022, −3.42927555869987635098179883476, −2.41712561681542491048625690657, −1.32595203114414613857570258152, 0,
1.32595203114414613857570258152, 2.41712561681542491048625690657, 3.42927555869987635098179883476, 4.08876702135675126351599181022, 4.79858245819537333446038366323, 5.86187005182510651321447425475, 6.41219830495361519372956586712, 7.21269514981927104076803165456, 7.80736164148569186936146869381
