| L(s) = 1 | + 1.30·2-s − 3-s − 0.285·4-s + 0.212·5-s − 1.30·6-s + 7-s − 2.99·8-s + 9-s + 0.278·10-s − 4.38·11-s + 0.285·12-s − 0.720·13-s + 1.30·14-s − 0.212·15-s − 3.34·16-s + 1.30·18-s − 2.07·19-s − 0.0608·20-s − 21-s − 5.74·22-s + 2.90·23-s + 2.99·24-s − 4.95·25-s − 0.942·26-s − 27-s − 0.285·28-s − 1.49·29-s + ⋯ |
| L(s) = 1 | + 0.925·2-s − 0.577·3-s − 0.142·4-s + 0.0951·5-s − 0.534·6-s + 0.377·7-s − 1.05·8-s + 0.333·9-s + 0.0881·10-s − 1.32·11-s + 0.0824·12-s − 0.199·13-s + 0.349·14-s − 0.0549·15-s − 0.836·16-s + 0.308·18-s − 0.475·19-s − 0.0135·20-s − 0.218·21-s − 1.22·22-s + 0.606·23-s + 0.610·24-s − 0.990·25-s − 0.184·26-s − 0.192·27-s − 0.0540·28-s − 0.277·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6069 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.533690423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.533690423\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 5 | \( 1 - 0.212T + 5T^{2} \) |
| 11 | \( 1 + 4.38T + 11T^{2} \) |
| 13 | \( 1 + 0.720T + 13T^{2} \) |
| 19 | \( 1 + 2.07T + 19T^{2} \) |
| 23 | \( 1 - 2.90T + 23T^{2} \) |
| 29 | \( 1 + 1.49T + 29T^{2} \) |
| 31 | \( 1 - 0.183T + 31T^{2} \) |
| 37 | \( 1 - 7.47T + 37T^{2} \) |
| 41 | \( 1 - 6.79T + 41T^{2} \) |
| 43 | \( 1 - 0.938T + 43T^{2} \) |
| 47 | \( 1 + 6.03T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 + 4.05T + 59T^{2} \) |
| 61 | \( 1 - 0.592T + 61T^{2} \) |
| 67 | \( 1 - 6.90T + 67T^{2} \) |
| 71 | \( 1 - 7.04T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 - 3.31T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 4.58T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−7.919487138388114779802347505278, −7.36249702634782518992214534003, −6.26295892837000524066867899624, −5.85390094503231678841406836319, −5.04925765470671188035956457294, −4.67435185724294973414810894815, −3.84382496321407237861772796946, −2.89725840712852807727721928563, −2.07491937895121823226562588249, −0.56013459899598547284982895856,
0.56013459899598547284982895856, 2.07491937895121823226562588249, 2.89725840712852807727721928563, 3.84382496321407237861772796946, 4.67435185724294973414810894815, 5.04925765470671188035956457294, 5.85390094503231678841406836319, 6.26295892837000524066867899624, 7.36249702634782518992214534003, 7.919487138388114779802347505278
