| L(s) = 1 | + 2.56·2-s − 3-s + 4.59·4-s − 1.61·5-s − 2.56·6-s − 7-s + 6.64·8-s + 9-s − 4.15·10-s − 3.75·11-s − 4.59·12-s − 3.22·13-s − 2.56·14-s + 1.61·15-s + 7.88·16-s + 5.51·17-s + 2.56·18-s + 7.99·19-s − 7.43·20-s + 21-s − 9.64·22-s + 3.68·23-s − 6.64·24-s − 2.37·25-s − 8.27·26-s − 27-s − 4.59·28-s + ⋯ |
| L(s) = 1 | + 1.81·2-s − 0.577·3-s + 2.29·4-s − 0.724·5-s − 1.04·6-s − 0.377·7-s + 2.35·8-s + 0.333·9-s − 1.31·10-s − 1.13·11-s − 1.32·12-s − 0.893·13-s − 0.686·14-s + 0.418·15-s + 1.97·16-s + 1.33·17-s + 0.605·18-s + 1.83·19-s − 1.66·20-s + 0.218·21-s − 2.05·22-s + 0.768·23-s − 1.35·24-s − 0.475·25-s − 1.62·26-s − 0.192·27-s − 0.867·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.294712663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.294712663\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 11 | \( 1 + 3.75T + 11T^{2} \) |
| 13 | \( 1 + 3.22T + 13T^{2} \) |
| 17 | \( 1 - 5.51T + 17T^{2} \) |
| 19 | \( 1 - 7.99T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 + 4.56T + 29T^{2} \) |
| 31 | \( 1 - 0.880T + 31T^{2} \) |
| 37 | \( 1 - 2.68T + 37T^{2} \) |
| 41 | \( 1 + 0.348T + 41T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 - 9.15T + 47T^{2} \) |
| 53 | \( 1 + 6.97T + 53T^{2} \) |
| 59 | \( 1 - 5.68T + 59T^{2} \) |
| 61 | \( 1 - 1.35T + 61T^{2} \) |
| 67 | \( 1 - 6.36T + 67T^{2} \) |
| 71 | \( 1 + 5.83T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 - 6.02T + 89T^{2} \) |
| 97 | \( 1 - 0.213T + 97T^{2} \) |
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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
−7.48101896101482707486051208684, −7.14914910798397262534383440018, −6.13272272219492082724240159456, −5.37100250701353727968035836831, −5.24740468999433571711198477572, −4.39741738846515253504617136790, −3.50572606208211631463293083670, −3.09718106896086472640419306420, −2.19700631976583663407938821172, −0.78341674475865222259631512187,
0.78341674475865222259631512187, 2.19700631976583663407938821172, 3.09718106896086472640419306420, 3.50572606208211631463293083670, 4.39741738846515253504617136790, 5.24740468999433571711198477572, 5.37100250701353727968035836831, 6.13272272219492082724240159456, 7.14914910798397262534383440018, 7.48101896101482707486051208684
