| L(s) = 1 | + 5-s − 2·7-s + 3·11-s − 4·13-s − 6·17-s + 7·19-s + 6·23-s + 25-s − 3·29-s − 5·31-s − 2·35-s − 4·37-s − 3·41-s − 8·43-s − 3·49-s + 6·53-s + 3·55-s − 3·59-s + 14·61-s − 4·65-s − 2·67-s + 15·71-s − 10·73-s − 6·77-s − 8·79-s − 6·85-s − 15·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 0.904·11-s − 1.10·13-s − 1.45·17-s + 1.60·19-s + 1.25·23-s + 1/5·25-s − 0.557·29-s − 0.898·31-s − 0.338·35-s − 0.657·37-s − 0.468·41-s − 1.21·43-s − 3/7·49-s + 0.824·53-s + 0.404·55-s − 0.390·59-s + 1.79·61-s − 0.496·65-s − 0.244·67-s + 1.78·71-s − 1.17·73-s − 0.683·77-s − 0.900·79-s − 0.650·85-s − 1.58·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 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 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.36910551268098437006354105682, −6.95580815465310481444364174346, −6.44141424309625544761448569689, −5.39327641900434759762671739275, −4.97606976074924679524544306440, −3.91315990274973933470108193110, −3.17936937765326174365210941161, −2.34094861958149271797270598426, −1.34219012430141233909477209709, 0,
1.34219012430141233909477209709, 2.34094861958149271797270598426, 3.17936937765326174365210941161, 3.91315990274973933470108193110, 4.97606976074924679524544306440, 5.39327641900434759762671739275, 6.44141424309625544761448569689, 6.95580815465310481444364174346, 7.36910551268098437006354105682
