| L(s) = 1 | − 3.15·3-s + 0.0614·5-s + 2.55·7-s + 6.95·9-s + 4.48·11-s − 1.07·13-s − 0.193·15-s − 3.56·17-s + 5.69·19-s − 8.05·21-s − 4.99·25-s − 12.4·27-s + 5.05·29-s + 9.56·31-s − 14.1·33-s + 0.156·35-s + 6.40·37-s + 3.38·39-s − 0.418·41-s − 1.35·43-s + 0.427·45-s + 9.28·47-s − 0.486·49-s + 11.2·51-s + 2.78·53-s + 0.275·55-s − 17.9·57-s + ⋯ |
| L(s) = 1 | − 1.82·3-s + 0.0274·5-s + 0.964·7-s + 2.31·9-s + 1.35·11-s − 0.297·13-s − 0.0500·15-s − 0.864·17-s + 1.30·19-s − 1.75·21-s − 0.999·25-s − 2.40·27-s + 0.939·29-s + 1.71·31-s − 2.46·33-s + 0.0265·35-s + 1.05·37-s + 0.542·39-s − 0.0653·41-s − 0.205·43-s + 0.0637·45-s + 1.35·47-s − 0.0695·49-s + 1.57·51-s + 0.382·53-s + 0.0371·55-s − 2.37·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.511410573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.511410573\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 3.15T + 3T^{2} \) |
| 5 | \( 1 - 0.0614T + 5T^{2} \) |
| 7 | \( 1 - 2.55T + 7T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 13 | \( 1 + 1.07T + 13T^{2} \) |
| 17 | \( 1 + 3.56T + 17T^{2} \) |
| 19 | \( 1 - 5.69T + 19T^{2} \) |
| 29 | \( 1 - 5.05T + 29T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 - 6.40T + 37T^{2} \) |
| 41 | \( 1 + 0.418T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 - 9.28T + 47T^{2} \) |
| 53 | \( 1 - 2.78T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 + 6.99T + 67T^{2} \) |
| 71 | \( 1 - 0.204T + 71T^{2} \) |
| 73 | \( 1 - 0.687T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 6.94T + 83T^{2} \) |
| 89 | \( 1 + 6.92T + 89T^{2} \) |
| 97 | \( 1 + 6.90T + 97T^{2} \) |
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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
−7.62799965085812020007806862322, −6.78416243145921051136301233401, −6.48917959021680623935594055238, −5.64594254423414855193888438440, −5.10808054605869984820306407051, −4.38726840627267748909093366100, −3.95220524795336299115663414177, −2.43828612376770112677420222661, −1.32840183237284230918655029157, −0.77663968201260552186367748987,
0.77663968201260552186367748987, 1.32840183237284230918655029157, 2.43828612376770112677420222661, 3.95220524795336299115663414177, 4.38726840627267748909093366100, 5.10808054605869984820306407051, 5.64594254423414855193888438440, 6.48917959021680623935594055238, 6.78416243145921051136301233401, 7.62799965085812020007806862322
