| L(s) = 1 | + 3-s + 5-s − 2·9-s − 5·11-s + 15-s − 3·17-s − 5·19-s − 3·23-s − 4·25-s − 5·27-s − 6·29-s − 3·31-s − 5·33-s + 3·37-s + 6·41-s − 8·43-s − 2·45-s − 9·47-s − 3·51-s − 5·53-s − 5·55-s − 5·57-s + 59-s − 7·61-s − 13·67-s − 3·69-s + 9·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 2/3·9-s − 1.50·11-s + 0.258·15-s − 0.727·17-s − 1.14·19-s − 0.625·23-s − 4/5·25-s − 0.962·27-s − 1.11·29-s − 0.538·31-s − 0.870·33-s + 0.493·37-s + 0.937·41-s − 1.21·43-s − 0.298·45-s − 1.31·47-s − 0.420·51-s − 0.686·53-s − 0.674·55-s − 0.662·57-s + 0.130·59-s − 0.896·61-s − 1.58·67-s − 0.361·69-s + 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 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
−14.69665787575518, −14.23391808666199, −13.53945889896543, −13.30226225459522, −12.90195046941748, −12.33259494240663, −11.57262862166748, −11.04539186417660, −10.78748582633515, −10.00082853912111, −9.683054487601149, −9.018457821406267, −8.574392040964465, −7.930496032382605, −7.764406657357359, −6.974084755408460, −6.158200218270673, −5.908186744232097, −5.245136391577475, −4.648148149298925, −3.966408308992286, −3.288486120652353, −2.638584866630334, −2.138631418387992, −1.660992264286757, 0, 0,
1.660992264286757, 2.138631418387992, 2.638584866630334, 3.288486120652353, 3.966408308992286, 4.648148149298925, 5.245136391577475, 5.908186744232097, 6.158200218270673, 6.974084755408460, 7.764406657357359, 7.930496032382605, 8.574392040964465, 9.018457821406267, 9.683054487601149, 10.00082853912111, 10.78748582633515, 11.04539186417660, 11.57262862166748, 12.33259494240663, 12.90195046941748, 13.30226225459522, 13.53945889896543, 14.23391808666199, 14.69665787575518
