| L(s) = 1 | + 3-s + 2.82·5-s + 4.24·7-s + 9-s − 4·11-s + 4.24·13-s + 2.82·15-s + 6·17-s − 2·19-s + 4.24·21-s + 2.82·23-s + 3.00·25-s + 27-s − 5.65·29-s − 4.24·31-s − 4·33-s + 12·35-s + 4.24·37-s + 4.24·39-s − 10·41-s + 6·43-s + 2.82·45-s − 2.82·47-s + 10.9·49-s + 6·51-s − 5.65·53-s − 11.3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.26·5-s + 1.60·7-s + 0.333·9-s − 1.20·11-s + 1.17·13-s + 0.730·15-s + 1.45·17-s − 0.458·19-s + 0.925·21-s + 0.589·23-s + 0.600·25-s + 0.192·27-s − 1.05·29-s − 0.762·31-s − 0.696·33-s + 2.02·35-s + 0.697·37-s + 0.679·39-s − 1.56·41-s + 0.914·43-s + 0.421·45-s − 0.412·47-s + 1.57·49-s + 0.840·51-s − 0.777·53-s − 1.52·55-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
| \(L(1)\) |
\(\approx\) |
\(3.680577813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.680577813\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 5.65T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 - 4.24T + 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 5.65T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 4.24T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 - 16T + 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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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
−8.635159546709504234667338907807, −7.974570990862250186745901337450, −7.51674537177975832348418944940, −6.32189295784899135292004637526, −5.42787690158571511922408982857, −5.14823138510250543370890819612, −3.95435075669513884682492463610, −2.91306687234717475144541813677, −1.92648141866716423382181490437, −1.33647034865909251669042359181,
1.33647034865909251669042359181, 1.92648141866716423382181490437, 2.91306687234717475144541813677, 3.95435075669513884682492463610, 5.14823138510250543370890819612, 5.42787690158571511922408982857, 6.32189295784899135292004637526, 7.51674537177975832348418944940, 7.974570990862250186745901337450, 8.635159546709504234667338907807
