| L(s) = 1 | + 34.8·2-s − 81·3-s + 700.·4-s − 913.·5-s − 2.82e3·6-s − 1.30e3·7-s + 6.56e3·8-s + 6.56e3·9-s − 3.18e4·10-s + 6.77e4·11-s − 5.67e4·12-s + 3.55e4·13-s − 4.54e4·14-s + 7.39e4·15-s − 1.30e5·16-s + 1.22e5·17-s + 2.28e5·18-s − 4.26e5·19-s − 6.39e5·20-s + 1.05e5·21-s + 2.36e6·22-s + 6.28e4·23-s − 5.31e5·24-s − 1.11e6·25-s + 1.23e6·26-s − 5.31e5·27-s − 9.14e5·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 0.577·3-s + 1.36·4-s − 0.653·5-s − 0.888·6-s − 0.205·7-s + 0.566·8-s + 0.333·9-s − 1.00·10-s + 1.39·11-s − 0.789·12-s + 0.345·13-s − 0.316·14-s + 0.377·15-s − 0.496·16-s + 0.356·17-s + 0.512·18-s − 0.750·19-s − 0.893·20-s + 0.118·21-s + 2.14·22-s + 0.0467·23-s − 0.326·24-s − 0.572·25-s + 0.531·26-s − 0.192·27-s − 0.281·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 81T \) |
| 59 | \( 1 + 1.21e7T \) |
| good | 2 | \( 1 - 34.8T + 512T^{2} \) |
| 5 | \( 1 + 913.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.30e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.77e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.55e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.22e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.26e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.28e4T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.02e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.48e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 6.71e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.26e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 5.20e5T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.65e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.11e7T + 3.29e15T^{2} \) |
| 61 | \( 1 - 6.62e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.85e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.02e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.05e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 4.56e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.38e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.61e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 5.78e8T + 7.60e17T^{2} \) |
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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
−11.09965979013041773981301494848, −9.695652979837882563732441304005, −8.306992720342373937995428053360, −6.77756883450743897541084608180, −6.25569716946456254709192508444, −5.00490612404437442767401272590, −4.06166750704303476630442869250, −3.31306124249898855011009179449, −1.60734881256800641892601713745, 0,
1.60734881256800641892601713745, 3.31306124249898855011009179449, 4.06166750704303476630442869250, 5.00490612404437442767401272590, 6.25569716946456254709192508444, 6.77756883450743897541084608180, 8.306992720342373937995428053360, 9.695652979837882563732441304005, 11.09965979013041773981301494848
