| L(s) = 1 | − 2-s + 1.67·3-s + 4-s − 2.32·5-s − 1.67·6-s − 2.00·7-s − 8-s − 0.203·9-s + 2.32·10-s − 4.47·11-s + 1.67·12-s + 6.47·13-s + 2.00·14-s − 3.88·15-s + 16-s − 4.47·17-s + 0.203·18-s + 7.95·19-s − 2.32·20-s − 3.36·21-s + 4.47·22-s − 23-s − 1.67·24-s + 0.406·25-s − 6.47·26-s − 5.35·27-s − 2.00·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.965·3-s + 0.5·4-s − 1.03·5-s − 0.682·6-s − 0.759·7-s − 0.353·8-s − 0.0677·9-s + 0.735·10-s − 1.35·11-s + 0.482·12-s + 1.79·13-s + 0.537·14-s − 1.00·15-s + 0.250·16-s − 1.08·17-s + 0.0479·18-s + 1.82·19-s − 0.519·20-s − 0.733·21-s + 0.955·22-s − 0.208·23-s − 0.341·24-s + 0.0812·25-s − 1.26·26-s − 1.03·27-s − 0.379·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9253272327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9253272327\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 - 1.67T + 3T^{2} \) |
| 5 | \( 1 + 2.32T + 5T^{2} \) |
| 7 | \( 1 + 2.00T + 7T^{2} \) |
| 11 | \( 1 + 4.47T + 11T^{2} \) |
| 13 | \( 1 - 6.47T + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 7.95T + 19T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 + 8.65T + 31T^{2} \) |
| 37 | \( 1 + 0.813T + 37T^{2} \) |
| 41 | \( 1 - 1.07T + 41T^{2} \) |
| 43 | \( 1 + 2.11T + 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 + 0.390T + 59T^{2} \) |
| 61 | \( 1 - 10.2T + 61T^{2} \) |
| 67 | \( 1 - 7.18T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 7.84T + 73T^{2} \) |
| 79 | \( 1 - 5.30T + 79T^{2} \) |
| 83 | \( 1 + 2.16T + 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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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
−8.193587655172013364226555590900, −7.44462685616220371855873996982, −7.16399942096175201481256389343, −5.93554782968468755441867015682, −5.43632897752707485142135810286, −3.97193568652623377620894618120, −3.51351790253161473623535099650, −2.87109791649773276079198545022, −1.91993050747811093945239393717, −0.51164349106109591653790710753,
0.51164349106109591653790710753, 1.91993050747811093945239393717, 2.87109791649773276079198545022, 3.51351790253161473623535099650, 3.97193568652623377620894618120, 5.43632897752707485142135810286, 5.93554782968468755441867015682, 7.16399942096175201481256389343, 7.44462685616220371855873996982, 8.193587655172013364226555590900
