| L(s) = 1 | + 4·7-s − 3·11-s + 4·13-s + 5·19-s − 6·23-s + 9·29-s + 5·31-s − 2·37-s + 9·41-s + 10·43-s − 6·47-s + 9·49-s − 12·53-s − 9·59-s − 10·61-s − 2·67-s − 3·71-s + 4·73-s − 12·77-s − 4·79-s + 6·83-s + 9·89-s + 16·91-s − 2·97-s + 15·101-s + 10·103-s + 6·107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.904·11-s + 1.10·13-s + 1.14·19-s − 1.25·23-s + 1.67·29-s + 0.898·31-s − 0.328·37-s + 1.40·41-s + 1.52·43-s − 0.875·47-s + 9/7·49-s − 1.64·53-s − 1.17·59-s − 1.28·61-s − 0.244·67-s − 0.356·71-s + 0.468·73-s − 1.36·77-s − 0.450·79-s + 0.658·83-s + 0.953·89-s + 1.67·91-s − 0.203·97-s + 1.49·101-s + 0.985·103-s + 0.580·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.736771232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.736771232\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.88262337731851311870417921589, −7.43465328810569102022993245115, −6.19813444562153059945956727549, −5.87078922534867975245864646985, −4.74850511007640324073209228514, −4.63150019637301685883305003179, −3.47040373748277996725817328454, −2.63796469638981412639676185280, −1.68310464171848719416573350918, −0.873605524273080761158972550445,
0.873605524273080761158972550445, 1.68310464171848719416573350918, 2.63796469638981412639676185280, 3.47040373748277996725817328454, 4.63150019637301685883305003179, 4.74850511007640324073209228514, 5.87078922534867975245864646985, 6.19813444562153059945956727549, 7.43465328810569102022993245115, 7.88262337731851311870417921589
