| L(s) = 1 | + 5-s + 7-s − 6·11-s + 13-s − 19-s + 6·23-s + 25-s + 6·29-s − 8·31-s + 35-s + 7·37-s + 6·41-s − 4·43-s + 12·47-s − 6·49-s − 6·53-s − 6·55-s − 11·61-s + 65-s − 7·67-s − 6·71-s + 11·73-s − 6·77-s + 79-s − 6·83-s + 12·89-s + 91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 1.80·11-s + 0.277·13-s − 0.229·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.169·35-s + 1.15·37-s + 0.937·41-s − 0.609·43-s + 1.75·47-s − 6/7·49-s − 0.824·53-s − 0.809·55-s − 1.40·61-s + 0.124·65-s − 0.855·67-s − 0.712·71-s + 1.28·73-s − 0.683·77-s + 0.112·79-s − 0.658·83-s + 1.27·89-s + 0.104·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.006796175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.006796175\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 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 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−7.65325310022281046164516713250, −7.30451912903569649949349756946, −6.25434577958309802736879966087, −5.71047071658936779529114408822, −4.95451858709415692382378827748, −4.50478637547779891832565542911, −3.26457319176422742677702276726, −2.66448701203453381941938096793, −1.83439279839273825075964346699, −0.68276771237936302501056002517,
0.68276771237936302501056002517, 1.83439279839273825075964346699, 2.66448701203453381941938096793, 3.26457319176422742677702276726, 4.50478637547779891832565542911, 4.95451858709415692382378827748, 5.71047071658936779529114408822, 6.25434577958309802736879966087, 7.30451912903569649949349756946, 7.65325310022281046164516713250
