| L(s) = 1 | − 0.381·2-s + 3-s − 1.85·4-s − 1.38·5-s − 0.381·6-s + 7-s + 1.47·8-s + 9-s + 0.527·10-s − 5.47·11-s − 1.85·12-s − 2.38·13-s − 0.381·14-s − 1.38·15-s + 3.14·16-s − 17-s − 0.381·18-s − 3·19-s + 2.56·20-s + 21-s + 2.09·22-s − 23-s + 1.47·24-s − 3.09·25-s + 0.909·26-s + 27-s − 1.85·28-s + ⋯ |
| L(s) = 1 | − 0.270·2-s + 0.577·3-s − 0.927·4-s − 0.618·5-s − 0.155·6-s + 0.377·7-s + 0.520·8-s + 0.333·9-s + 0.166·10-s − 1.64·11-s − 0.535·12-s − 0.660·13-s − 0.102·14-s − 0.356·15-s + 0.786·16-s − 0.242·17-s − 0.0900·18-s − 0.688·19-s + 0.572·20-s + 0.218·21-s + 0.445·22-s − 0.208·23-s + 0.300·24-s − 0.618·25-s + 0.178·26-s + 0.192·27-s − 0.350·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 11 | \( 1 + 5.47T + 11T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 29 | \( 1 + 7.47T + 29T^{2} \) |
| 31 | \( 1 + 3.76T + 31T^{2} \) |
| 37 | \( 1 - 1.47T + 37T^{2} \) |
| 41 | \( 1 + 4.70T + 41T^{2} \) |
| 43 | \( 1 - 8.09T + 43T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 + 3.38T + 53T^{2} \) |
| 59 | \( 1 + 6.14T + 59T^{2} \) |
| 61 | \( 1 - 13.7T + 61T^{2} \) |
| 67 | \( 1 - 4.14T + 67T^{2} \) |
| 71 | \( 1 + 3.90T + 71T^{2} \) |
| 73 | \( 1 - 2.70T + 73T^{2} \) |
| 79 | \( 1 - 0.527T + 79T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 - 3.14T + 89T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−10.39823081149817439046632654717, −9.578180479679418414146930744942, −8.639076508177464065000002641327, −7.87526213274663625327873522313, −7.40093200900812343193993508106, −5.57563154157219914651930065485, −4.66043629966342784974621871937, −3.69656774805218198534431793330, −2.23088730330174361858629621432, 0,
2.23088730330174361858629621432, 3.69656774805218198534431793330, 4.66043629966342784974621871937, 5.57563154157219914651930065485, 7.40093200900812343193993508106, 7.87526213274663625327873522313, 8.639076508177464065000002641327, 9.578180479679418414146930744942, 10.39823081149817439046632654717
