| L(s) = 1 | − 9.23·2-s + 53.2·4-s − 89.5·5-s − 121.·7-s − 195.·8-s + 826.·10-s − 542.·11-s − 202.·13-s + 1.12e3·14-s + 105.·16-s + 431.·17-s + 362.·19-s − 4.76e3·20-s + 5.00e3·22-s − 1.69e3·23-s + 4.89e3·25-s + 1.86e3·26-s − 6.46e3·28-s + 7.67e3·29-s − 2.42e3·31-s + 5.29e3·32-s − 3.98e3·34-s + 1.08e4·35-s + 2.53e3·37-s − 3.34e3·38-s + 1.75e4·40-s − 6.56e3·41-s + ⋯ |
| L(s) = 1 | − 1.63·2-s + 1.66·4-s − 1.60·5-s − 0.936·7-s − 1.08·8-s + 2.61·10-s − 1.35·11-s − 0.332·13-s + 1.52·14-s + 0.102·16-s + 0.361·17-s + 0.230·19-s − 2.66·20-s + 2.20·22-s − 0.667·23-s + 1.56·25-s + 0.541·26-s − 1.55·28-s + 1.69·29-s − 0.453·31-s + 0.914·32-s − 0.590·34-s + 1.50·35-s + 0.304·37-s − 0.375·38-s + 1.73·40-s − 0.610·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 2 | \( 1 + 9.23T + 32T^{2} \) |
| 5 | \( 1 + 89.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 121.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 542.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 202.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 431.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 362.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.42e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.53e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.56e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 779.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.38e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.45e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 1.96e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.61e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.71e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.20e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.36e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.30e4T + 8.58e9T^{2} \) |
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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
−9.647903687345790878194543527044, −8.561833022433470759241078591201, −7.910536058351025065081247203572, −7.38124256349623457237942577970, −6.41610071972192113120315075416, −4.85431576183670031043323573819, −3.49200259488376302888539535732, −2.49323669158266087095324403618, −0.72796773310621760945971895548, 0,
0.72796773310621760945971895548, 2.49323669158266087095324403618, 3.49200259488376302888539535732, 4.85431576183670031043323573819, 6.41610071972192113120315075416, 7.38124256349623457237942577970, 7.910536058351025065081247203572, 8.561833022433470759241078591201, 9.647903687345790878194543527044
