| L(s) = 1 | − 5.44·2-s + 0.176·3-s + 21.6·4-s − 13.0·5-s − 0.961·6-s + 7·7-s − 74.1·8-s − 26.9·9-s + 71.0·10-s + 11·11-s + 3.81·12-s + 66.1·13-s − 38.1·14-s − 2.30·15-s + 230.·16-s + 119.·17-s + 146.·18-s − 20.7·19-s − 282.·20-s + 1.23·21-s − 59.8·22-s + 121.·23-s − 13.0·24-s + 45.3·25-s − 360.·26-s − 9.53·27-s + 151.·28-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 0.0339·3-s + 2.70·4-s − 1.16·5-s − 0.0654·6-s + 0.377·7-s − 3.27·8-s − 0.998·9-s + 2.24·10-s + 0.301·11-s + 0.0918·12-s + 1.41·13-s − 0.727·14-s − 0.0396·15-s + 3.60·16-s + 1.70·17-s + 1.92·18-s − 0.250·19-s − 3.15·20-s + 0.0128·21-s − 0.580·22-s + 1.10·23-s − 0.111·24-s + 0.363·25-s − 2.71·26-s − 0.0679·27-s + 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5329315007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5329315007\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 5.44T + 8T^{2} \) |
| 3 | \( 1 - 0.176T + 27T^{2} \) |
| 5 | \( 1 + 13.0T + 125T^{2} \) |
| 13 | \( 1 - 66.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 173.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 106.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 122.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 209.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 87.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 292.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 205.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 738.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 457.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 69.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 760.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.27e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 143.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 292.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 504.T + 9.12e5T^{2} \) |
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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
−14.44833324054901186766565012936, −12.20852629167252702742167257465, −11.35605593911138281638589241525, −10.71576383732345563952455562635, −9.169761866753312704582788935087, −8.288255405981257210661816043349, −7.59272422741030994383735960735, −6.08610877114800903690189110681, −3.23310482669984967489602321297, −0.944651254519751412762112291892,
0.944651254519751412762112291892, 3.23310482669984967489602321297, 6.08610877114800903690189110681, 7.59272422741030994383735960735, 8.288255405981257210661816043349, 9.169761866753312704582788935087, 10.71576383732345563952455562635, 11.35605593911138281638589241525, 12.20852629167252702742167257465, 14.44833324054901186766565012936
