| L(s) = 1 | + 2.91·3-s + 1.91·5-s + 3.81·7-s + 5.49·9-s − 0.0492·11-s + 2.32·13-s + 5.59·15-s − 6.76·17-s − 8.12·19-s + 11.1·21-s + 6.65·23-s − 1.31·25-s + 7.27·27-s + 9.38·29-s − 2.64·31-s − 0.143·33-s + 7.32·35-s − 0.740·37-s + 6.78·39-s − 10.7·41-s + 10.8·43-s + 10.5·45-s − 8.22·47-s + 7.55·49-s − 19.7·51-s + 8.63·53-s − 0.0946·55-s + ⋯ |
| L(s) = 1 | + 1.68·3-s + 0.858·5-s + 1.44·7-s + 1.83·9-s − 0.0148·11-s + 0.645·13-s + 1.44·15-s − 1.63·17-s − 1.86·19-s + 2.42·21-s + 1.38·23-s − 0.262·25-s + 1.40·27-s + 1.74·29-s − 0.475·31-s − 0.0250·33-s + 1.23·35-s − 0.121·37-s + 1.08·39-s − 1.67·41-s + 1.66·43-s + 1.57·45-s − 1.19·47-s + 1.07·49-s − 2.75·51-s + 1.18·53-s − 0.0127·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.023667064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.023667064\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 2.91T + 3T^{2} \) |
| 5 | \( 1 - 1.91T + 5T^{2} \) |
| 7 | \( 1 - 3.81T + 7T^{2} \) |
| 11 | \( 1 + 0.0492T + 11T^{2} \) |
| 13 | \( 1 - 2.32T + 13T^{2} \) |
| 17 | \( 1 + 6.76T + 17T^{2} \) |
| 19 | \( 1 + 8.12T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 - 9.38T + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 + 0.740T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 10.8T + 43T^{2} \) |
| 47 | \( 1 + 8.22T + 47T^{2} \) |
| 53 | \( 1 - 8.63T + 53T^{2} \) |
| 59 | \( 1 - 12.7T + 59T^{2} \) |
| 61 | \( 1 - 6.76T + 61T^{2} \) |
| 67 | \( 1 + 3.07T + 67T^{2} \) |
| 71 | \( 1 + 5.82T + 71T^{2} \) |
| 73 | \( 1 - 8.06T + 73T^{2} \) |
| 79 | \( 1 - 3.75T + 79T^{2} \) |
| 83 | \( 1 + 4.63T + 83T^{2} \) |
| 89 | \( 1 - 6.74T + 89T^{2} \) |
| 97 | \( 1 + 9.93T + 97T^{2} \) |
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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
−8.547181651396057787381320973349, −8.114990017044989130003133952561, −6.97787545643593831358726291170, −6.52172000559484171316225049967, −5.28512213344599328970900272494, −4.48361246580185196286478162229, −3.89402173956055491070956637088, −2.60358258452137417348079547477, −2.15493161549598002311536958131, −1.38210371623341449884948606473,
1.38210371623341449884948606473, 2.15493161549598002311536958131, 2.60358258452137417348079547477, 3.89402173956055491070956637088, 4.48361246580185196286478162229, 5.28512213344599328970900272494, 6.52172000559484171316225049967, 6.97787545643593831358726291170, 8.114990017044989130003133952561, 8.547181651396057787381320973349
