| L(s) = 1 | + 9·3-s + 106.·5-s + 49·7-s + 81·9-s − 250.·11-s − 300.·13-s + 957.·15-s + 2.02e3·17-s − 2.25e3·19-s + 441·21-s + 3.09e3·23-s + 8.19e3·25-s + 729·27-s − 6.60e3·29-s + 833.·31-s − 2.25e3·33-s + 5.21e3·35-s + 8.95e3·37-s − 2.70e3·39-s − 7.20e3·41-s + 1.44e4·43-s + 8.61e3·45-s − 1.79e4·47-s + 2.40e3·49-s + 1.81e4·51-s − 1.58e4·53-s − 2.66e4·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.90·5-s + 0.377·7-s + 0.333·9-s − 0.623·11-s − 0.493·13-s + 1.09·15-s + 1.69·17-s − 1.43·19-s + 0.218·21-s + 1.21·23-s + 2.62·25-s + 0.192·27-s − 1.45·29-s + 0.155·31-s − 0.359·33-s + 0.719·35-s + 1.07·37-s − 0.284·39-s − 0.669·41-s + 1.19·43-s + 0.634·45-s − 1.18·47-s + 0.142·49-s + 0.979·51-s − 0.773·53-s − 1.18·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.990911079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.990911079\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 7 | \( 1 - 49T \) |
| good | 5 | \( 1 - 106.T + 3.12e3T^{2} \) |
| 11 | \( 1 + 250.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 300.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.02e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.25e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.09e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.60e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 833.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 8.95e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.20e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.44e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.79e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.58e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.67e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.67e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.44e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.04e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.87e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.72e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.06e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.81e4T + 8.58e9T^{2} \) |
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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
−13.30371829236956142135149667851, −12.66447960539359114067315176878, −10.77972170603551267427942591296, −9.889446047578646413826003908938, −9.016983762345606754708982675844, −7.62200588550972514227763524971, −6.09872618699120467039700897528, −4.99941570222576614436449275870, −2.79822937856559072442082209064, −1.58768645208602917670190450878,
1.58768645208602917670190450878, 2.79822937856559072442082209064, 4.99941570222576614436449275870, 6.09872618699120467039700897528, 7.62200588550972514227763524971, 9.016983762345606754708982675844, 9.889446047578646413826003908938, 10.77972170603551267427942591296, 12.66447960539359114067315176878, 13.30371829236956142135149667851
