| L(s) = 1 | + 2-s + 0.527·3-s + 4-s + 5-s + 0.527·6-s − 1.45·7-s + 8-s − 2.72·9-s + 10-s − 2.62·11-s + 0.527·12-s + 1.24·13-s − 1.45·14-s + 0.527·15-s + 16-s + 2.45·17-s − 2.72·18-s − 1.20·19-s + 20-s − 0.768·21-s − 2.62·22-s − 1.35·23-s + 0.527·24-s + 25-s + 1.24·26-s − 3.01·27-s − 1.45·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.304·3-s + 0.5·4-s + 0.447·5-s + 0.215·6-s − 0.550·7-s + 0.353·8-s − 0.907·9-s + 0.316·10-s − 0.790·11-s + 0.152·12-s + 0.344·13-s − 0.389·14-s + 0.136·15-s + 0.250·16-s + 0.596·17-s − 0.641·18-s − 0.276·19-s + 0.223·20-s − 0.167·21-s − 0.558·22-s − 0.282·23-s + 0.107·24-s + 0.200·25-s + 0.243·26-s − 0.581·27-s − 0.275·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 0.527T + 3T^{2} \) |
| 7 | \( 1 + 1.45T + 7T^{2} \) |
| 11 | \( 1 + 2.62T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 - 2.45T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 - 0.521T + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 4.79T + 41T^{2} \) |
| 43 | \( 1 + 6.82T + 43T^{2} \) |
| 47 | \( 1 - 7.76T + 47T^{2} \) |
| 53 | \( 1 - 2.85T + 53T^{2} \) |
| 59 | \( 1 - 4.64T + 59T^{2} \) |
| 61 | \( 1 + 5.90T + 61T^{2} \) |
| 67 | \( 1 + 2.08T + 67T^{2} \) |
| 71 | \( 1 - 9.96T + 71T^{2} \) |
| 73 | \( 1 + 5.44T + 73T^{2} \) |
| 79 | \( 1 + 9.71T + 79T^{2} \) |
| 83 | \( 1 + 17.4T + 83T^{2} \) |
| 89 | \( 1 - 5.36T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−7.66638311383741721753648462013, −6.89187263305276328057551086786, −6.16622583253131947123235023377, −5.51743519678368758750229640697, −5.04236885821256279367241968855, −3.86599129981886267847614435739, −3.21847513165867978101153912966, −2.56615207187570350302756839160, −1.62991744800477580005902871612, 0,
1.62991744800477580005902871612, 2.56615207187570350302756839160, 3.21847513165867978101153912966, 3.86599129981886267847614435739, 5.04236885821256279367241968855, 5.51743519678368758750229640697, 6.16622583253131947123235023377, 6.89187263305276328057551086786, 7.66638311383741721753648462013
