| L(s) = 1 | − 14.8·2-s − 239.·3-s − 292.·4-s + 625·5-s + 3.55e3·6-s + 2.40e3·7-s + 1.19e4·8-s + 3.77e4·9-s − 9.26e3·10-s − 1.65e4·11-s + 7.00e4·12-s + 2.63e4·13-s − 3.56e4·14-s − 1.49e5·15-s − 2.72e4·16-s − 1.44e5·17-s − 5.60e5·18-s − 1.59e5·19-s − 1.82e5·20-s − 5.75e5·21-s + 2.45e5·22-s + 2.07e6·23-s − 2.85e6·24-s + 3.90e5·25-s − 3.90e5·26-s − 4.34e6·27-s − 7.01e5·28-s + ⋯ |
| L(s) = 1 | − 0.655·2-s − 1.70·3-s − 0.570·4-s + 0.447·5-s + 1.11·6-s + 0.377·7-s + 1.02·8-s + 1.91·9-s − 0.293·10-s − 0.340·11-s + 0.974·12-s + 0.255·13-s − 0.247·14-s − 0.764·15-s − 0.103·16-s − 0.418·17-s − 1.25·18-s − 0.281·19-s − 0.255·20-s − 0.645·21-s + 0.222·22-s + 1.54·23-s − 1.75·24-s + 0.200·25-s − 0.167·26-s − 1.57·27-s − 0.215·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 + 14.8T + 512T^{2} \) |
| 3 | \( 1 + 239.T + 1.96e4T^{2} \) |
| 11 | \( 1 + 1.65e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 2.63e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.44e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.59e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.07e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.94e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 4.22e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.29e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.87e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.54e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.95e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.31e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.68e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 6.70e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.56e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 6.95e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 7.83e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.26e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.31e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.14e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.46e9T + 7.60e17T^{2} \) |
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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
−13.64664731359557892116212425239, −12.55884841704641277353379114775, −11.09525134376827389948380142627, −10.37175410722683179387072686485, −8.964318341000569705781051570471, −7.19428124837091048835379462598, −5.65037960346970989466098281220, −4.59274530100514490928847016055, −1.30518867075844682515039124990, 0,
1.30518867075844682515039124990, 4.59274530100514490928847016055, 5.65037960346970989466098281220, 7.19428124837091048835379462598, 8.964318341000569705781051570471, 10.37175410722683179387072686485, 11.09525134376827389948380142627, 12.55884841704641277353379114775, 13.64664731359557892116212425239
