| L(s) = 1 | − 2.02e7·2-s − 2.01e10·3-s + 2.71e14·4-s − 3.11e16·5-s + 4.09e17·6-s − 1.26e20·7-s − 2.64e21·8-s − 2.61e22·9-s + 6.32e23·10-s − 1.78e23·11-s − 5.46e24·12-s − 1.12e26·13-s + 2.56e27·14-s + 6.28e26·15-s + 1.55e28·16-s + 4.31e28·17-s + 5.31e29·18-s − 7.19e29·19-s − 8.45e30·20-s + 2.55e30·21-s + 3.62e30·22-s + 8.88e30·23-s + 5.34e31·24-s + 2.60e32·25-s + 2.28e33·26-s + 1.06e33·27-s − 3.43e34·28-s + ⋯ |
| L(s) = 1 | − 1.71·2-s − 0.123·3-s + 1.92·4-s − 1.16·5-s + 0.211·6-s − 1.74·7-s − 1.58·8-s − 0.984·9-s + 2.00·10-s − 0.0601·11-s − 0.238·12-s − 0.746·13-s + 2.99·14-s + 0.144·15-s + 0.787·16-s + 0.523·17-s + 1.68·18-s − 0.640·19-s − 2.25·20-s + 0.216·21-s + 0.102·22-s + 0.0886·23-s + 0.196·24-s + 0.367·25-s + 1.27·26-s + 0.245·27-s − 3.36·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(48-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+47/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(24)\) |
\(\approx\) |
\(0.1274431610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1274431610\) |
| \(L(\frac{49}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 2.02e7T + 1.40e14T^{2} \) |
| 3 | \( 1 + 2.01e10T + 2.65e22T^{2} \) |
| 5 | \( 1 + 3.11e16T + 7.10e32T^{2} \) |
| 7 | \( 1 + 1.26e20T + 5.24e39T^{2} \) |
| 11 | \( 1 + 1.78e23T + 8.81e48T^{2} \) |
| 13 | \( 1 + 1.12e26T + 2.26e52T^{2} \) |
| 17 | \( 1 - 4.31e28T + 6.77e57T^{2} \) |
| 19 | \( 1 + 7.19e29T + 1.26e60T^{2} \) |
| 23 | \( 1 - 8.88e30T + 1.00e64T^{2} \) |
| 29 | \( 1 - 2.84e34T + 5.40e68T^{2} \) |
| 31 | \( 1 - 1.11e35T + 1.24e70T^{2} \) |
| 37 | \( 1 + 9.87e36T + 5.07e73T^{2} \) |
| 41 | \( 1 + 3.56e37T + 6.32e75T^{2} \) |
| 43 | \( 1 + 3.37e38T + 5.92e76T^{2} \) |
| 47 | \( 1 + 1.22e39T + 3.87e78T^{2} \) |
| 53 | \( 1 + 1.08e39T + 1.09e81T^{2} \) |
| 59 | \( 1 + 1.45e40T + 1.69e83T^{2} \) |
| 61 | \( 1 - 4.79e41T + 8.13e83T^{2} \) |
| 67 | \( 1 - 3.35e42T + 6.69e85T^{2} \) |
| 71 | \( 1 - 9.20e42T + 1.02e87T^{2} \) |
| 73 | \( 1 + 5.22e43T + 3.76e87T^{2} \) |
| 79 | \( 1 + 7.68e43T + 1.54e89T^{2} \) |
| 83 | \( 1 + 1.71e45T + 1.57e90T^{2} \) |
| 89 | \( 1 + 9.70e45T + 4.18e91T^{2} \) |
| 97 | \( 1 - 5.35e46T + 2.38e93T^{2} \) |
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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
−19.78880197808427816036450246665, −19.15513542619149578736597245464, −16.96309481192772037816863607953, −15.73424110202675186102707082064, −11.94815917211133557240383180920, −10.07922282019894674193145975946, −8.415174341120272693876340904649, −6.75012880687945371681079481690, −2.99795514297973204710885199718, −0.32132544407402366206956330567,
0.32132544407402366206956330567, 2.99795514297973204710885199718, 6.75012880687945371681079481690, 8.415174341120272693876340904649, 10.07922282019894674193145975946, 11.94815917211133557240383180920, 15.73424110202675186102707082064, 16.96309481192772037816863607953, 19.15513542619149578736597245464, 19.78880197808427816036450246665
