| L(s) = 1 | + 3-s + 9-s − 4·13-s + 4·19-s + 8·23-s + 27-s − 2·29-s + 4·31-s − 2·37-s − 4·39-s + 10·41-s + 43-s − 12·47-s − 7·49-s + 2·53-s + 4·57-s + 8·59-s − 12·61-s − 8·67-s + 8·69-s − 12·71-s + 6·73-s + 81-s − 6·83-s − 2·87-s + 12·89-s + 4·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.10·13-s + 0.917·19-s + 1.66·23-s + 0.192·27-s − 0.371·29-s + 0.718·31-s − 0.328·37-s − 0.640·39-s + 1.56·41-s + 0.152·43-s − 1.75·47-s − 49-s + 0.274·53-s + 0.529·57-s + 1.04·59-s − 1.53·61-s − 0.977·67-s + 0.963·69-s − 1.42·71-s + 0.702·73-s + 1/9·81-s − 0.658·83-s − 0.214·87-s + 1.27·89-s + 0.414·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.036757915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.036757915\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.15424040985416, −12.58005186405755, −12.17715863866774, −11.63503280205314, −11.17442450520177, −10.71586354274463, −10.06102102337204, −9.695805144457663, −9.303800078745092, −8.842043020809664, −8.300002058735417, −7.692263194174867, −7.381179672107588, −6.951970639664170, −6.353658995832028, −5.734674832606814, −5.113991358983634, −4.714378229149364, −4.274440394491342, −3.383165852705409, −3.043782485461324, −2.593706779380467, −1.836641220353364, −1.227097330948176, −0.4845432412780815,
0.4845432412780815, 1.227097330948176, 1.836641220353364, 2.593706779380467, 3.043782485461324, 3.383165852705409, 4.274440394491342, 4.714378229149364, 5.113991358983634, 5.734674832606814, 6.353658995832028, 6.951970639664170, 7.381179672107588, 7.692263194174867, 8.300002058735417, 8.842043020809664, 9.303800078745092, 9.695805144457663, 10.06102102337204, 10.71586354274463, 11.17442450520177, 11.63503280205314, 12.17715863866774, 12.58005186405755, 13.15424040985416
