| L(s) = 1 | − 16.1·2-s − 48.3·3-s + 133.·4-s − 272.·5-s + 780.·6-s + 1.10e3·7-s − 81.7·8-s + 148.·9-s + 4.40e3·10-s + 685.·11-s − 6.42e3·12-s + 1.32e4·13-s − 1.77e4·14-s + 1.31e4·15-s − 1.57e4·16-s − 1.15e3·17-s − 2.39e3·18-s − 1.29e4·19-s − 3.63e4·20-s − 5.32e4·21-s − 1.10e4·22-s − 1.06e5·23-s + 3.94e3·24-s − 3.63e3·25-s − 2.13e5·26-s + 9.85e4·27-s + 1.46e5·28-s + ⋯ |
| L(s) = 1 | − 1.42·2-s − 1.03·3-s + 1.03·4-s − 0.976·5-s + 1.47·6-s + 1.21·7-s − 0.0564·8-s + 0.0677·9-s + 1.39·10-s + 0.155·11-s − 1.07·12-s + 1.66·13-s − 1.73·14-s + 1.00·15-s − 0.958·16-s − 0.0570·17-s − 0.0966·18-s − 0.433·19-s − 1.01·20-s − 1.25·21-s − 0.221·22-s − 1.83·23-s + 0.0582·24-s − 0.0464·25-s − 2.38·26-s + 0.963·27-s + 1.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 - 7.95e4T \) |
| good | 2 | \( 1 + 16.1T + 128T^{2} \) |
| 3 | \( 1 + 48.3T + 2.18e3T^{2} \) |
| 5 | \( 1 + 272.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.10e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 685.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.32e4T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.15e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.29e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 1.06e5T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.14e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 3.45e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 7.35e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 7.14e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 9.15e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 575.T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.55e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.66e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.03e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.58e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.26e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 7.66e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.26e7T + 8.07e13T^{2} \) |
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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
−13.96609857819406390462297178478, −11.89300415138124070475039985190, −11.27903326424657361550394725260, −10.43167810473048490506021469891, −8.548904058447261788140538795977, −7.937189985566166330894510636149, −6.25195478361896149052804814142, −4.38407571352645288159798361556, −1.35380385121915730978728954136, 0,
1.35380385121915730978728954136, 4.38407571352645288159798361556, 6.25195478361896149052804814142, 7.937189985566166330894510636149, 8.548904058447261788140538795977, 10.43167810473048490506021469891, 11.27903326424657361550394725260, 11.89300415138124070475039985190, 13.96609857819406390462297178478
