| L(s) = 1 | + 189.·3-s − 729.·5-s − 2.40e3·7-s + 1.61e4·9-s − 3.34e4·11-s − 1.31e5·13-s − 1.38e5·15-s − 5.04e5·17-s + 2.80e5·19-s − 4.54e5·21-s + 1.37e6·23-s − 1.42e6·25-s − 6.70e5·27-s + 2.14e6·29-s + 2.31e6·31-s − 6.32e6·33-s + 1.75e6·35-s − 2.44e6·37-s − 2.49e7·39-s − 3.40e7·41-s − 1.28e7·43-s − 1.17e7·45-s − 2.22e7·47-s + 5.76e6·49-s − 9.54e7·51-s + 4.76e7·53-s + 2.44e7·55-s + ⋯ |
| L(s) = 1 | + 1.34·3-s − 0.522·5-s − 0.377·7-s + 0.819·9-s − 0.688·11-s − 1.28·13-s − 0.704·15-s − 1.46·17-s + 0.492·19-s − 0.509·21-s + 1.02·23-s − 0.727·25-s − 0.242·27-s + 0.563·29-s + 0.450·31-s − 0.928·33-s + 0.197·35-s − 0.214·37-s − 1.72·39-s − 1.88·41-s − 0.573·43-s − 0.428·45-s − 0.663·47-s + 0.142·49-s − 1.97·51-s + 0.830·53-s + 0.359·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 2.40e3T \) |
| good | 3 | \( 1 - 189.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 729.T + 1.95e6T^{2} \) |
| 11 | \( 1 + 3.34e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.31e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.04e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 2.80e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.37e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.14e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.31e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 2.44e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.40e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.28e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.22e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.76e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.04e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.53e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.10e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.00e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.73e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.59e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.02e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.91e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.97e8T + 7.60e17T^{2} \) |
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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
−13.03837742796310815364330427980, −11.71128867607423890126690001971, −10.15659174999226919968439429932, −9.053544037250333776475004945493, −8.014783233017976394429587599810, −6.95597219647352061561149595434, −4.81319446127923288418302911128, −3.29536195782328239556843341376, −2.25453664697253233675057329534, 0,
2.25453664697253233675057329534, 3.29536195782328239556843341376, 4.81319446127923288418302911128, 6.95597219647352061561149595434, 8.014783233017976394429587599810, 9.053544037250333776475004945493, 10.15659174999226919968439429932, 11.71128867607423890126690001971, 13.03837742796310815364330427980
