| L(s) = 1 | + 10.2·2-s − 225.·3-s − 407.·4-s − 1.61e3·5-s − 2.29e3·6-s − 263.·7-s − 9.38e3·8-s + 3.10e4·9-s − 1.65e4·10-s − 1.58e4·11-s + 9.18e4·12-s + 3.72e4·13-s − 2.68e3·14-s + 3.65e5·15-s + 1.12e5·16-s − 2.49e5·17-s + 3.17e5·18-s − 7.35e4·19-s + 6.60e5·20-s + 5.92e4·21-s − 1.61e5·22-s − 1.69e6·23-s + 2.11e6·24-s + 6.71e5·25-s + 3.79e5·26-s − 2.56e6·27-s + 1.07e5·28-s + ⋯ |
| L(s) = 1 | + 0.451·2-s − 1.60·3-s − 0.796·4-s − 1.15·5-s − 0.724·6-s − 0.0414·7-s − 0.810·8-s + 1.57·9-s − 0.522·10-s − 0.326·11-s + 1.27·12-s + 0.361·13-s − 0.0186·14-s + 1.86·15-s + 0.430·16-s − 0.725·17-s + 0.712·18-s − 0.129·19-s + 0.923·20-s + 0.0665·21-s − 0.147·22-s − 1.26·23-s + 1.30·24-s + 0.343·25-s + 0.163·26-s − 0.930·27-s + 0.0329·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.3939293524\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3939293524\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 - 7.07e5T \) |
| good | 2 | \( 1 - 10.2T + 512T^{2} \) |
| 3 | \( 1 + 225.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.61e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 263.T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.58e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 3.72e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.49e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.35e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.69e6T + 1.80e12T^{2} \) |
| 31 | \( 1 - 8.52e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.01e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.51e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.94e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.04e8T + 3.29e15T^{2} \) |
| 59 | \( 1 - 4.40e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.29e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.03e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 2.30e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.68e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.14e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.43e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 2.06e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 9.20e7T + 7.60e17T^{2} \) |
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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
−15.30990783248942550036091333329, −13.55177777056949240379811080961, −12.27221540480859579298386346168, −11.58304974207824718694147561273, −10.22842509764974296885872232761, −8.269999903892563632013045150967, −6.45305029965102699008210842666, −5.05629840409174647294806186437, −3.96186386385503726868127382581, −0.45676259305982058228727354207,
0.45676259305982058228727354207, 3.96186386385503726868127382581, 5.05629840409174647294806186437, 6.45305029965102699008210842666, 8.269999903892563632013045150967, 10.22842509764974296885872232761, 11.58304974207824718694147561273, 12.27221540480859579298386346168, 13.55177777056949240379811080961, 15.30990783248942550036091333329
