| L(s) = 1 | + 1.26·2-s − 2.62·3-s − 0.398·4-s − 2.90·5-s − 3.32·6-s − 3.03·8-s + 3.90·9-s − 3.67·10-s + 2.03·11-s + 1.04·12-s + 13-s + 7.62·15-s − 3.04·16-s + 3.99·17-s + 4.93·18-s + 6.96·19-s + 1.15·20-s + 2.57·22-s − 0.627·23-s + 7.97·24-s + 3.42·25-s + 1.26·26-s − 2.37·27-s + 1.09·29-s + 9.65·30-s − 10.4·31-s + 2.21·32-s + ⋯ |
| L(s) = 1 | + 0.894·2-s − 1.51·3-s − 0.199·4-s − 1.29·5-s − 1.35·6-s − 1.07·8-s + 1.30·9-s − 1.16·10-s + 0.614·11-s + 0.302·12-s + 0.277·13-s + 1.96·15-s − 0.761·16-s + 0.969·17-s + 1.16·18-s + 1.59·19-s + 0.258·20-s + 0.549·22-s − 0.130·23-s + 1.62·24-s + 0.685·25-s + 0.248·26-s − 0.456·27-s + 0.203·29-s + 1.76·30-s − 1.87·31-s + 0.391·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8488849415\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8488849415\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 1.26T + 2T^{2} \) |
| 3 | \( 1 + 2.62T + 3T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 17 | \( 1 - 3.99T + 17T^{2} \) |
| 19 | \( 1 - 6.96T + 19T^{2} \) |
| 23 | \( 1 + 0.627T + 23T^{2} \) |
| 29 | \( 1 - 1.09T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 3.08T + 37T^{2} \) |
| 41 | \( 1 + 0.521T + 41T^{2} \) |
| 43 | \( 1 - 0.329T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 7.11T + 53T^{2} \) |
| 59 | \( 1 - 2.03T + 59T^{2} \) |
| 61 | \( 1 - 2.40T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 3.60T + 71T^{2} \) |
| 73 | \( 1 - 2.97T + 73T^{2} \) |
| 79 | \( 1 + 8.76T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 + 2.32T + 97T^{2} \) |
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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
−11.04710187344569022810355330165, −9.903102968238379156622246185888, −8.870415460217353437468124255515, −7.64760966936838998623177031553, −6.82347048110939850676750561411, −5.65871437213022781620982719046, −5.22160992529478884201996075507, −4.07052869479064403728805460215, −3.46232434429809137797318091278, −0.75742506270143291825187926934,
0.75742506270143291825187926934, 3.46232434429809137797318091278, 4.07052869479064403728805460215, 5.22160992529478884201996075507, 5.65871437213022781620982719046, 6.82347048110939850676750561411, 7.64760966936838998623177031553, 8.870415460217353437468124255515, 9.903102968238379156622246185888, 11.04710187344569022810355330165
