| L(s) = 1 | + 3·5-s − 7-s − 6·11-s − 4·13-s + 3·17-s − 2·19-s + 6·23-s + 4·25-s − 6·29-s + 4·31-s − 3·35-s − 7·37-s − 3·41-s + 43-s − 9·47-s + 49-s − 6·53-s − 18·55-s − 9·59-s − 10·61-s − 12·65-s + 4·67-s + 2·73-s + 6·77-s + 79-s − 3·83-s + 9·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 1.80·11-s − 1.10·13-s + 0.727·17-s − 0.458·19-s + 1.25·23-s + 4/5·25-s − 1.11·29-s + 0.718·31-s − 0.507·35-s − 1.15·37-s − 0.468·41-s + 0.152·43-s − 1.31·47-s + 1/7·49-s − 0.824·53-s − 2.42·55-s − 1.17·59-s − 1.28·61-s − 1.48·65-s + 0.488·67-s + 0.234·73-s + 0.683·77-s + 0.112·79-s − 0.329·83-s + 0.976·85-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 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
−8.311270091027571152283912289413, −7.58077569587831207824696562961, −6.83048798134075240948185763985, −5.94905967912697497452429576264, −5.24156886937486665769086126481, −4.81781072476011914834397859806, −3.22194024007524560609322005234, −2.59559695408065534572876809133, −1.69139831568062347701224305803, 0,
1.69139831568062347701224305803, 2.59559695408065534572876809133, 3.22194024007524560609322005234, 4.81781072476011914834397859806, 5.24156886937486665769086126481, 5.94905967912697497452429576264, 6.83048798134075240948185763985, 7.58077569587831207824696562961, 8.311270091027571152283912289413
