| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 2·13-s + 14-s − 16-s − 2·17-s − 4·19-s − 2·26-s − 28-s + 6·29-s + 5·32-s − 2·34-s − 6·37-s − 4·38-s − 6·41-s − 4·43-s + 49-s + 2·52-s − 2·53-s − 3·56-s + 6·58-s − 4·59-s − 6·61-s + 7·64-s − 12·67-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.392·26-s − 0.188·28-s + 1.11·29-s + 0.883·32-s − 0.342·34-s − 0.986·37-s − 0.648·38-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.277·52-s − 0.274·53-s − 0.400·56-s + 0.787·58-s − 0.520·59-s − 0.768·61-s + 7/8·64-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5692343220\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5692343220\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.29319553899973, −12.46077554794384, −12.35000145621259, −11.94012798536155, −11.27811252460345, −10.81925194098739, −10.25937259150522, −9.872144955867895, −9.247060241945792, −8.807844750287872, −8.344509951102485, −8.017399540156727, −7.210179125053364, −6.723421133966218, −6.289262588013212, −5.700856082599447, −5.090036162974811, −4.777479332062149, −4.280660233411301, −3.798903419450861, −3.068156874708034, −2.666109985472084, −1.897510851959664, −1.254647851246648, −0.1910694254212603,
0.1910694254212603, 1.254647851246648, 1.897510851959664, 2.666109985472084, 3.068156874708034, 3.798903419450861, 4.280660233411301, 4.777479332062149, 5.090036162974811, 5.700856082599447, 6.289262588013212, 6.723421133966218, 7.210179125053364, 8.017399540156727, 8.344509951102485, 8.807844750287872, 9.247060241945792, 9.872144955867895, 10.25937259150522, 10.81925194098739, 11.27811252460345, 11.94012798536155, 12.35000145621259, 12.46077554794384, 13.29319553899973
