| L(s) = 1 | + 5-s + 4·7-s − 3·11-s − 4·13-s + 3·17-s − 5·19-s − 6·23-s + 25-s + 6·29-s − 2·31-s + 4·35-s − 4·37-s − 3·41-s − 11·43-s + 9·49-s + 6·53-s − 3·55-s + 3·59-s − 10·61-s − 4·65-s − 5·67-s − 6·71-s − 7·73-s − 12·77-s − 14·79-s − 12·83-s + 3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 0.904·11-s − 1.10·13-s + 0.727·17-s − 1.14·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.359·31-s + 0.676·35-s − 0.657·37-s − 0.468·41-s − 1.67·43-s + 9/7·49-s + 0.824·53-s − 0.404·55-s + 0.390·59-s − 1.28·61-s − 0.496·65-s − 0.610·67-s − 0.712·71-s − 0.819·73-s − 1.36·77-s − 1.57·79-s − 1.31·83-s + 0.325·85-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 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−7.70905508262901120226408682192, −7.12916834275611027363476095787, −6.14336902356744622503824521582, −5.40197073686164486106906564979, −4.84689739783531494793997369270, −4.26087392994872042224529875995, −3.01886073458577638061403975201, −2.16182316817130746863273688365, −1.54806662506126606640986996169, 0,
1.54806662506126606640986996169, 2.16182316817130746863273688365, 3.01886073458577638061403975201, 4.26087392994872042224529875995, 4.84689739783531494793997369270, 5.40197073686164486106906564979, 6.14336902356744622503824521582, 7.12916834275611027363476095787, 7.70905508262901120226408682192
