| L(s) = 1 | + 33.5·2-s + 86.6·3-s + 610.·4-s − 2.61e3·5-s + 2.90e3·6-s + 8.62e3·7-s + 3.29e3·8-s − 1.21e4·9-s − 8.76e4·10-s + 3.40e4·11-s + 5.28e4·12-s + 4.80e3·13-s + 2.88e5·14-s − 2.26e5·15-s − 2.02e5·16-s − 1.73e5·17-s − 4.07e5·18-s − 4.73e5·19-s − 1.59e6·20-s + 7.46e5·21-s + 1.14e6·22-s − 1.73e6·23-s + 2.85e5·24-s + 4.89e6·25-s + 1.61e5·26-s − 2.76e6·27-s + 5.26e6·28-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 0.617·3-s + 1.19·4-s − 1.87·5-s + 0.914·6-s + 1.35·7-s + 0.284·8-s − 0.618·9-s − 2.77·10-s + 0.701·11-s + 0.736·12-s + 0.0466·13-s + 2.00·14-s − 1.15·15-s − 0.771·16-s − 0.504·17-s − 0.916·18-s − 0.832·19-s − 2.23·20-s + 0.837·21-s + 1.03·22-s − 1.29·23-s + 0.175·24-s + 2.50·25-s + 0.0691·26-s − 0.999·27-s + 1.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + 1.41e8T \) |
| good | 2 | \( 1 - 33.5T + 512T^{2} \) |
| 3 | \( 1 - 86.6T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.61e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.62e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.40e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.80e3T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.73e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.73e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.73e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.85e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 8.74e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.41e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 9.24e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.69e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.72e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 7.97e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.01e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.12e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 5.87e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.98e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.26e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.52e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.33e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.03e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.02e9T + 7.60e17T^{2} \) |
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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
−11.62688304702804957343138377675, −11.07033072828992492129541705999, −8.698294690595666926855759470560, −8.097177999277049220071341751246, −6.86389986352395892576990963227, −5.23566882915024778895801910502, −4.11650970987348381897354339046, −3.62344896913874970032805969598, −2.10239971240852981930236268830, 0,
2.10239971240852981930236268830, 3.62344896913874970032805969598, 4.11650970987348381897354339046, 5.23566882915024778895801910502, 6.86389986352395892576990963227, 8.097177999277049220071341751246, 8.698294690595666926855759470560, 11.07033072828992492129541705999, 11.62688304702804957343138377675
