| L(s) = 1 | − 38.2·2-s − 81·3-s + 949.·4-s + 625·5-s + 3.09e3·6-s + 2.40e3·7-s − 1.67e4·8-s + 6.56e3·9-s − 2.38e4·10-s + 4.05e4·11-s − 7.69e4·12-s − 3.27e4·13-s − 9.17e4·14-s − 5.06e4·15-s + 1.53e5·16-s − 5.20e5·17-s − 2.50e5·18-s − 5.19e5·19-s + 5.93e5·20-s − 1.94e5·21-s − 1.55e6·22-s + 7.45e5·23-s + 1.35e6·24-s + 3.90e5·25-s + 1.25e6·26-s − 5.31e5·27-s + 2.27e6·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s − 0.577·3-s + 1.85·4-s + 0.447·5-s + 0.975·6-s + 0.377·7-s − 1.44·8-s + 0.333·9-s − 0.755·10-s + 0.835·11-s − 1.07·12-s − 0.317·13-s − 0.638·14-s − 0.258·15-s + 0.584·16-s − 1.51·17-s − 0.563·18-s − 0.913·19-s + 0.829·20-s − 0.218·21-s − 1.41·22-s + 0.555·23-s + 0.833·24-s + 0.200·25-s + 0.536·26-s − 0.192·27-s + 0.700·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 81T \) |
| 5 | \( 1 - 625T \) |
| 7 | \( 1 - 2.40e3T \) |
| good | 2 | \( 1 + 38.2T + 512T^{2} \) |
| 11 | \( 1 - 4.05e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 3.27e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 5.20e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.19e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 7.45e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.80e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.91e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.95e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.54e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.17e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.61e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.11e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.89e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.80e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 6.60e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.44e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 2.04e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.39e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.98e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.94e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.22e8T + 7.60e17T^{2} \) |
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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
−11.08364644976722190615750548594, −10.31261438288241028958616546826, −9.223764759778297997129570157734, −8.458496756352975908274367314594, −7.05065131223665426914380586676, −6.30167108342216378586000453329, −4.56603411669057536681134032591, −2.29880992150977619848151006470, −1.22948254764012627743126428457, 0,
1.22948254764012627743126428457, 2.29880992150977619848151006470, 4.56603411669057536681134032591, 6.30167108342216378586000453329, 7.05065131223665426914380586676, 8.458496756352975908274367314594, 9.223764759778297997129570157734, 10.31261438288241028958616546826, 11.08364644976722190615750548594
