| L(s) = 1 | − 3-s − 0.473·5-s − 4.55·7-s + 9-s − 3.49·11-s − 0.0840·13-s + 0.473·15-s − 3.61·17-s − 3.61·19-s + 4.55·21-s − 2.82·23-s − 4.77·25-s − 27-s + 7.30·29-s + 0.557·31-s + 3.49·33-s + 2.15·35-s + 6.20·37-s + 0.0840·39-s − 9.27·41-s − 2.27·43-s − 0.473·45-s + 2.82·47-s + 13.7·49-s + 3.61·51-s + 0.697·53-s + 1.65·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.211·5-s − 1.72·7-s + 0.333·9-s − 1.05·11-s − 0.0233·13-s + 0.122·15-s − 0.877·17-s − 0.829·19-s + 0.994·21-s − 0.589·23-s − 0.955·25-s − 0.192·27-s + 1.35·29-s + 0.100·31-s + 0.608·33-s + 0.364·35-s + 1.01·37-s + 0.0134·39-s − 1.44·41-s − 0.347·43-s − 0.0706·45-s + 0.412·47-s + 1.96·49-s + 0.506·51-s + 0.0958·53-s + 0.223·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4411116808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4411116808\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 + 0.473T + 5T^{2} \) |
| 7 | \( 1 + 4.55T + 7T^{2} \) |
| 11 | \( 1 + 3.49T + 11T^{2} \) |
| 13 | \( 1 + 0.0840T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 + 3.61T + 19T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 - 7.30T + 29T^{2} \) |
| 31 | \( 1 - 0.557T + 31T^{2} \) |
| 37 | \( 1 - 6.20T + 37T^{2} \) |
| 41 | \( 1 + 9.27T + 41T^{2} \) |
| 43 | \( 1 + 2.27T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 0.697T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 - 5.33T + 67T^{2} \) |
| 71 | \( 1 - 9.11T + 71T^{2} \) |
| 73 | \( 1 + 0.541T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−8.691570209726364465816336509650, −7.951796059363429919239668160432, −7.00638938704548957779032535845, −6.38764093318650211638703888602, −5.87907232179393807089773081520, −4.83147318091589642998273263721, −4.02043268620955719180630533188, −3.07501098781709785572695988197, −2.19585836955146979442182775257, −0.38558982328734312295465831128,
0.38558982328734312295465831128, 2.19585836955146979442182775257, 3.07501098781709785572695988197, 4.02043268620955719180630533188, 4.83147318091589642998273263721, 5.87907232179393807089773081520, 6.38764093318650211638703888602, 7.00638938704548957779032535845, 7.951796059363429919239668160432, 8.691570209726364465816336509650
