| L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 4·11-s + 13-s − 15-s − 17-s + 2·21-s + 8·23-s + 25-s − 27-s + 6·29-s − 2·31-s − 4·33-s − 2·35-s + 4·37-s − 39-s − 2·41-s − 4·43-s + 45-s + 2·47-s − 3·49-s + 51-s + 10·53-s + 4·55-s + 10·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s − 0.242·17-s + 0.436·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.359·31-s − 0.696·33-s − 0.338·35-s + 0.657·37-s − 0.160·39-s − 0.312·41-s − 0.609·43-s + 0.149·45-s + 0.291·47-s − 3/7·49-s + 0.140·51-s + 1.37·53-s + 0.539·55-s + 1.30·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.277552861\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.277552861\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−15.21683053214332, −14.81090503666504, −14.27664250544505, −13.48799886854325, −13.19382664496725, −12.68149488394296, −11.98358506852943, −11.57922152660426, −11.02175196095840, −10.33882811943776, −9.939731103763380, −9.216519654904430, −8.915049342388356, −8.235729434560115, −7.235701125383244, −6.731440497499156, −6.477224797720531, −5.741845107882129, −5.144878875148142, −4.443828974515086, −3.748382739514268, −3.094069580659687, −2.277331399007005, −1.285669749669311, −0.6876124982637155,
0.6876124982637155, 1.285669749669311, 2.277331399007005, 3.094069580659687, 3.748382739514268, 4.443828974515086, 5.144878875148142, 5.741845107882129, 6.477224797720531, 6.731440497499156, 7.235701125383244, 8.235729434560115, 8.915049342388356, 9.216519654904430, 9.939731103763380, 10.33882811943776, 11.02175196095840, 11.57922152660426, 11.98358506852943, 12.68149488394296, 13.19382664496725, 13.48799886854325, 14.27664250544505, 14.81090503666504, 15.21683053214332
