L(s) = 1 | + 3-s + 0.732·5-s + 2·7-s + 9-s + 11-s − 13-s + 0.732·15-s + 6.73·17-s + 0.535·19-s + 2·21-s + 2·23-s − 4.46·25-s + 27-s − 4.19·29-s + 8.19·31-s + 33-s + 1.46·35-s − 2·37-s − 39-s − 5.46·41-s + 6.19·43-s + 0.732·45-s + 1.46·47-s − 3·49-s + 6.73·51-s + 2·53-s + 0.732·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.327·5-s + 0.755·7-s + 0.333·9-s + 0.301·11-s − 0.277·13-s + 0.189·15-s + 1.63·17-s + 0.122·19-s + 0.436·21-s + 0.417·23-s − 0.892·25-s + 0.192·27-s − 0.779·29-s + 1.47·31-s + 0.174·33-s + 0.247·35-s − 0.328·37-s − 0.160·39-s − 0.853·41-s + 0.944·43-s + 0.109·45-s + 0.213·47-s − 0.428·49-s + 0.942·51-s + 0.274·53-s + 0.0987·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.378044845\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.378044845\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 - 0.732T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 - 0.535T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 - 8.19T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 - 6.19T + 43T^{2} \) |
| 47 | \( 1 - 1.46T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 - 2.92T + 73T^{2} \) |
| 79 | \( 1 - 3.66T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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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
−8.047194347599458755185533529774, −7.37904211150250074669923576502, −6.69657231921138206498571001853, −5.67936037556233281986047736690, −5.23581747150699622756025849721, −4.28657349903500893686903193050, −3.56849747613032293681101630727, −2.69986103811844792744796893861, −1.82052726405432040227438496091, −0.976174301898693071832387706033,
0.976174301898693071832387706033, 1.82052726405432040227438496091, 2.69986103811844792744796893861, 3.56849747613032293681101630727, 4.28657349903500893686903193050, 5.23581747150699622756025849721, 5.67936037556233281986047736690, 6.69657231921138206498571001853, 7.37904211150250074669923576502, 8.047194347599458755185533529774