| L(s) = 1 | − 106.·3-s + 1.30e3·5-s − 9.19e3·7-s − 8.34e3·9-s − 6.22e4·11-s + 1.41e5·13-s − 1.38e5·15-s + 8.35e4·17-s + 9.41e5·19-s + 9.79e5·21-s − 5.68e5·23-s − 2.52e5·25-s + 2.98e6·27-s − 1.83e6·29-s + 7.34e6·31-s + 6.62e6·33-s − 1.19e7·35-s − 8.01e6·37-s − 1.51e7·39-s + 1.95e7·41-s − 3.46e7·43-s − 1.08e7·45-s + 5.63e7·47-s + 4.42e7·49-s − 8.89e6·51-s − 3.28e7·53-s − 8.11e7·55-s + ⋯ |
| L(s) = 1 | − 0.758·3-s + 0.933·5-s − 1.44·7-s − 0.424·9-s − 1.28·11-s + 1.37·13-s − 0.708·15-s + 0.242·17-s + 1.65·19-s + 1.09·21-s − 0.423·23-s − 0.129·25-s + 1.08·27-s − 0.482·29-s + 1.42·31-s + 0.972·33-s − 1.35·35-s − 0.703·37-s − 1.04·39-s + 1.08·41-s − 1.54·43-s − 0.395·45-s + 1.68·47-s + 1.09·49-s − 0.184·51-s − 0.572·53-s − 1.19·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 17 | \( 1 - 8.35e4T \) |
| good | 3 | \( 1 + 106.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.30e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 9.19e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 6.22e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.41e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 9.41e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 5.68e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.83e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.34e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 8.01e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.95e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.46e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.63e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 3.28e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.04e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 5.69e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.58e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 9.81e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 6.27e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.38e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.69e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.21e7T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.11e8T + 7.60e17T^{2} \) |
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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
−9.982090061703300093367892329299, −9.122523481184465397144662734685, −7.87577273812866126756082292397, −6.51561311989483520219377214791, −5.88248712231227860934440114813, −5.24449727595454114294211346504, −3.48000997860403807407443639319, −2.61481942892958420318349175278, −1.05392109524173819567724885004, 0,
1.05392109524173819567724885004, 2.61481942892958420318349175278, 3.48000997860403807407443639319, 5.24449727595454114294211346504, 5.88248712231227860934440114813, 6.51561311989483520219377214791, 7.87577273812866126756082292397, 9.122523481184465397144662734685, 9.982090061703300093367892329299
