| L(s) = 1 | + 3-s + 5-s + 9-s − 4·11-s − 6·13-s + 15-s − 2·17-s − 4·19-s − 8·23-s + 25-s + 27-s − 6·29-s − 4·33-s + 2·37-s − 6·39-s + 10·41-s − 4·43-s + 45-s − 7·49-s − 2·51-s + 10·53-s − 4·55-s − 4·57-s + 12·59-s + 2·61-s − 6·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.258·15-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.696·33-s + 0.328·37-s − 0.960·39-s + 1.56·41-s − 0.609·43-s + 0.149·45-s − 49-s − 0.280·51-s + 1.37·53-s − 0.539·55-s − 0.529·57-s + 1.56·59-s + 0.256·61-s − 0.744·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.14056582666577, −12.89322953305367, −12.29298783590903, −11.86152079733500, −11.30290257637799, −10.66149962203496, −10.25313297365105, −9.979958182575162, −9.380190389096095, −9.098744801165186, −8.350009526994096, −7.868511745178220, −7.721551268622858, −6.972471391947521, −6.619483383739759, −5.864083003112406, −5.456286193093223, −4.960048841033619, −4.315197227358362, −3.978757823073138, −3.166153489635427, −2.494675490458593, −2.183279811828093, −1.889243504491958, −0.6501158656978175, 0,
0.6501158656978175, 1.889243504491958, 2.183279811828093, 2.494675490458593, 3.166153489635427, 3.978757823073138, 4.315197227358362, 4.960048841033619, 5.456286193093223, 5.864083003112406, 6.619483383739759, 6.972471391947521, 7.721551268622858, 7.868511745178220, 8.350009526994096, 9.098744801165186, 9.380190389096095, 9.979958182575162, 10.25313297365105, 10.66149962203496, 11.30290257637799, 11.86152079733500, 12.29298783590903, 12.89322953305367, 13.14056582666577
