| L(s) = 1 | − 0.718·2-s − 2.25·3-s − 7.48·4-s − 17.9·5-s + 1.62·6-s + 7·7-s + 11.1·8-s − 21.9·9-s + 12.9·10-s − 18.3·11-s + 16.8·12-s + 62.8·13-s − 5.02·14-s + 40.5·15-s + 51.8·16-s − 22.1·17-s + 15.7·18-s + 23.1·19-s + 134.·20-s − 15.7·21-s + 13.2·22-s + 23·23-s − 25.0·24-s + 198.·25-s − 45.1·26-s + 110.·27-s − 52.3·28-s + ⋯ |
| L(s) = 1 | − 0.253·2-s − 0.434·3-s − 0.935·4-s − 1.60·5-s + 0.110·6-s + 0.377·7-s + 0.491·8-s − 0.811·9-s + 0.408·10-s − 0.504·11-s + 0.406·12-s + 1.34·13-s − 0.0959·14-s + 0.697·15-s + 0.810·16-s − 0.315·17-s + 0.206·18-s + 0.279·19-s + 1.50·20-s − 0.164·21-s + 0.128·22-s + 0.208·23-s − 0.213·24-s + 1.58·25-s − 0.340·26-s + 0.786·27-s − 0.353·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5452234432\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5452234432\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 + 0.718T + 8T^{2} \) |
| 3 | \( 1 + 2.25T + 27T^{2} \) |
| 5 | \( 1 + 17.9T + 125T^{2} \) |
| 11 | \( 1 + 18.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 22.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 23.1T + 6.85e3T^{2} \) |
| 29 | \( 1 + 35.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 194.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 189.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 4.81T + 6.89e4T^{2} \) |
| 43 | \( 1 - 334.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 45.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 633.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 322.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 862.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 123.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 295.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 837.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 680.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 775.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 553.T + 9.12e5T^{2} \) |
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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
−12.28864733582075269489976128562, −11.24999644075501168322399834895, −10.76354185750352613944966923823, −9.037976033452633717203804361019, −8.314935846677551915251924564242, −7.49179265647439203562452539387, −5.75422193670831789985825096235, −4.53952906360669166081486165063, −3.48509466213116998353271520730, −0.62651610420456844602603929975,
0.62651610420456844602603929975, 3.48509466213116998353271520730, 4.53952906360669166081486165063, 5.75422193670831789985825096235, 7.49179265647439203562452539387, 8.314935846677551915251924564242, 9.037976033452633717203804361019, 10.76354185750352613944966923823, 11.24999644075501168322399834895, 12.28864733582075269489976128562
