| L(s) = 1 | − 5-s + 7-s + 6·11-s + 4·13-s + 3·17-s − 2·19-s − 2·23-s − 4·25-s + 6·29-s − 4·31-s − 35-s + 5·37-s + 5·41-s − 9·43-s + 5·47-s + 49-s − 6·53-s − 6·55-s − 7·59-s + 14·61-s − 4·65-s − 12·67-s + 8·71-s − 10·73-s + 6·77-s − 5·79-s + 11·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 1.80·11-s + 1.10·13-s + 0.727·17-s − 0.458·19-s − 0.417·23-s − 4/5·25-s + 1.11·29-s − 0.718·31-s − 0.169·35-s + 0.821·37-s + 0.780·41-s − 1.37·43-s + 0.729·47-s + 1/7·49-s − 0.824·53-s − 0.809·55-s − 0.911·59-s + 1.79·61-s − 0.496·65-s − 1.46·67-s + 0.949·71-s − 1.17·73-s + 0.683·77-s − 0.562·79-s + 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.141735091\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.141735091\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.681289058396699958698087924764, −8.058837105797167750622819473131, −7.25858432480085412397353380158, −6.32971854675632218825125453660, −5.91562599090996976542691916246, −4.65454805256349914811237503090, −3.95348670580962397077701253696, −3.34597982563495596655332115594, −1.87256837126608301063986327859, −0.962722642780806718863586408717,
0.962722642780806718863586408717, 1.87256837126608301063986327859, 3.34597982563495596655332115594, 3.95348670580962397077701253696, 4.65454805256349914811237503090, 5.91562599090996976542691916246, 6.32971854675632218825125453660, 7.25858432480085412397353380158, 8.058837105797167750622819473131, 8.681289058396699958698087924764
