| L(s) = 1 | + 2.47·2-s + 3-s + 4.14·4-s − 1.24·5-s + 2.47·6-s − 1.48·7-s + 5.32·8-s + 9-s − 3.09·10-s + 2.18·11-s + 4.14·12-s + 4.58·13-s − 3.68·14-s − 1.24·15-s + 4.90·16-s + 2.22·17-s + 2.47·18-s − 3.59·19-s − 5.17·20-s − 1.48·21-s + 5.42·22-s − 1.04·23-s + 5.32·24-s − 3.44·25-s + 11.3·26-s + 27-s − 6.15·28-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 0.577·3-s + 2.07·4-s − 0.558·5-s + 1.01·6-s − 0.561·7-s + 1.88·8-s + 0.333·9-s − 0.979·10-s + 0.660·11-s + 1.19·12-s + 1.27·13-s − 0.983·14-s − 0.322·15-s + 1.22·16-s + 0.539·17-s + 0.584·18-s − 0.825·19-s − 1.15·20-s − 0.323·21-s + 1.15·22-s − 0.218·23-s + 1.08·24-s − 0.688·25-s + 2.22·26-s + 0.192·27-s − 1.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 669 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.416079445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.416079445\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 223 | \( 1 + T \) |
| good | 2 | \( 1 - 2.47T + 2T^{2} \) |
| 5 | \( 1 + 1.24T + 5T^{2} \) |
| 7 | \( 1 + 1.48T + 7T^{2} \) |
| 11 | \( 1 - 2.18T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 - 2.22T + 17T^{2} \) |
| 19 | \( 1 + 3.59T + 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 + 5.04T + 29T^{2} \) |
| 31 | \( 1 - 1.24T + 31T^{2} \) |
| 37 | \( 1 + 1.96T + 37T^{2} \) |
| 41 | \( 1 + 7.81T + 41T^{2} \) |
| 43 | \( 1 + 0.397T + 43T^{2} \) |
| 47 | \( 1 - 0.351T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 6.79T + 59T^{2} \) |
| 61 | \( 1 - 6.49T + 61T^{2} \) |
| 67 | \( 1 - 10.0T + 67T^{2} \) |
| 71 | \( 1 - 2.85T + 71T^{2} \) |
| 73 | \( 1 - 5.59T + 73T^{2} \) |
| 79 | \( 1 - 0.736T + 79T^{2} \) |
| 83 | \( 1 - 7.48T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−10.91279625923514044095389878596, −9.748115883128089288519737753335, −8.630642331867785116520319740149, −7.67222609089644360375345103507, −6.58560869601001338238359622170, −6.04588700359142001901296898692, −4.78161628518519911503301527232, −3.64463899795004228013136285256, −3.48897797028078267031600663272, −1.90719937191492144851607794074,
1.90719937191492144851607794074, 3.48897797028078267031600663272, 3.64463899795004228013136285256, 4.78161628518519911503301527232, 6.04588700359142001901296898692, 6.58560869601001338238359622170, 7.67222609089644360375345103507, 8.630642331867785116520319740149, 9.748115883128089288519737753335, 10.91279625923514044095389878596
