| L(s) = 1 | + 2.24·2-s − 3-s + 3.04·4-s − 4.04·5-s − 2.24·6-s − 1.80·7-s + 2.35·8-s + 9-s − 9.09·10-s − 11-s − 3.04·12-s − 4.13·13-s − 4.04·14-s + 4.04·15-s − 0.801·16-s − 3.35·17-s + 2.24·18-s − 19-s − 12.3·20-s + 1.80·21-s − 2.24·22-s + 7.31·23-s − 2.35·24-s + 11.3·25-s − 9.29·26-s − 27-s − 5.49·28-s + ⋯ |
| L(s) = 1 | + 1.58·2-s − 0.577·3-s + 1.52·4-s − 1.81·5-s − 0.917·6-s − 0.681·7-s + 0.833·8-s + 0.333·9-s − 2.87·10-s − 0.301·11-s − 0.880·12-s − 1.14·13-s − 1.08·14-s + 1.04·15-s − 0.200·16-s − 0.814·17-s + 0.529·18-s − 0.229·19-s − 2.76·20-s + 0.393·21-s − 0.479·22-s + 1.52·23-s − 0.481·24-s + 2.27·25-s − 1.82·26-s − 0.192·27-s − 1.03·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.24T + 2T^{2} \) |
| 5 | \( 1 + 4.04T + 5T^{2} \) |
| 7 | \( 1 + 1.80T + 7T^{2} \) |
| 13 | \( 1 + 4.13T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 23 | \( 1 - 7.31T + 23T^{2} \) |
| 29 | \( 1 - 8.00T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 + 4.50T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 12.4T + 43T^{2} \) |
| 47 | \( 1 + 4.02T + 47T^{2} \) |
| 53 | \( 1 + 10.6T + 53T^{2} \) |
| 59 | \( 1 - 6.51T + 59T^{2} \) |
| 61 | \( 1 + 4.66T + 61T^{2} \) |
| 67 | \( 1 - 6.80T + 67T^{2} \) |
| 71 | \( 1 - 5.03T + 71T^{2} \) |
| 73 | \( 1 - 14.2T + 73T^{2} \) |
| 79 | \( 1 + 3.13T + 79T^{2} \) |
| 83 | \( 1 - 3.82T + 83T^{2} \) |
| 89 | \( 1 + 6.99T + 89T^{2} \) |
| 97 | \( 1 + 3.25T + 97T^{2} \) |
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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
−10.78651454765059016202610138905, −9.354676318847034742134288127395, −8.103577223997899842040001601549, −6.98756806927920459087440570584, −6.66335968344755607130141174577, −5.13631363302847026162632205350, −4.64494150598860412244688752899, −3.66273869984981939087818941652, −2.79510487889179487783268245210, 0,
2.79510487889179487783268245210, 3.66273869984981939087818941652, 4.64494150598860412244688752899, 5.13631363302847026162632205350, 6.66335968344755607130141174577, 6.98756806927920459087440570584, 8.103577223997899842040001601549, 9.354676318847034742134288127395, 10.78651454765059016202610138905
