| L(s) = 1 | − 2.80·2-s − 1.28·3-s + 5.84·4-s − 1.26·5-s + 3.60·6-s − 2.24·7-s − 10.7·8-s − 1.34·9-s + 3.55·10-s − 3.60·11-s − 7.52·12-s − 0.255·13-s + 6.28·14-s + 1.63·15-s + 18.4·16-s − 6.73·17-s + 3.75·18-s + 7.52·19-s − 7.40·20-s + 2.88·21-s + 10.1·22-s + 7.98·23-s + 13.8·24-s − 3.39·25-s + 0.715·26-s + 5.59·27-s − 13.1·28-s + ⋯ |
| L(s) = 1 | − 1.98·2-s − 0.743·3-s + 2.92·4-s − 0.566·5-s + 1.47·6-s − 0.848·7-s − 3.80·8-s − 0.447·9-s + 1.12·10-s − 1.08·11-s − 2.17·12-s − 0.0708·13-s + 1.68·14-s + 0.421·15-s + 4.61·16-s − 1.63·17-s + 0.886·18-s + 1.72·19-s − 1.65·20-s + 0.630·21-s + 2.15·22-s + 1.66·23-s + 2.83·24-s − 0.678·25-s + 0.140·26-s + 1.07·27-s − 2.47·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01764039174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01764039174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + T \) |
| 139 | \( 1 - T \) |
| good | 2 | \( 1 + 2.80T + 2T^{2} \) |
| 3 | \( 1 + 1.28T + 3T^{2} \) |
| 5 | \( 1 + 1.26T + 5T^{2} \) |
| 7 | \( 1 + 2.24T + 7T^{2} \) |
| 11 | \( 1 + 3.60T + 11T^{2} \) |
| 13 | \( 1 + 0.255T + 13T^{2} \) |
| 17 | \( 1 + 6.73T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 - 7.98T + 23T^{2} \) |
| 31 | \( 1 + 0.488T + 31T^{2} \) |
| 37 | \( 1 + 5.31T + 37T^{2} \) |
| 41 | \( 1 + 9.66T + 41T^{2} \) |
| 43 | \( 1 + 6.73T + 43T^{2} \) |
| 47 | \( 1 + 9.53T + 47T^{2} \) |
| 53 | \( 1 - 1.47T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 - 5.52T + 61T^{2} \) |
| 67 | \( 1 + 15.6T + 67T^{2} \) |
| 71 | \( 1 - 0.700T + 71T^{2} \) |
| 73 | \( 1 + 6.68T + 73T^{2} \) |
| 79 | \( 1 - 3.27T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + 5.88T + 89T^{2} \) |
| 97 | \( 1 + 3.42T + 97T^{2} \) |
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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
−8.493880095982171798206437374783, −7.85726125226850133332392749027, −6.94075572733173327225995714028, −6.76881775960480328240153382751, −5.73348660302232004783979048209, −5.04057140869801609097643362507, −3.21797793620447798273616175383, −2.85852224532284204838841937829, −1.52135287206267018938497769554, −0.10476848173284679134134963767,
0.10476848173284679134134963767, 1.52135287206267018938497769554, 2.85852224532284204838841937829, 3.21797793620447798273616175383, 5.04057140869801609097643362507, 5.73348660302232004783979048209, 6.76881775960480328240153382751, 6.94075572733173327225995714028, 7.85726125226850133332392749027, 8.493880095982171798206437374783
