| L(s) = 1 | − 32·4-s − 57·5-s + 1.02e3·16-s + 1.82e3·20-s − 981·23-s + 124·25-s − 7.77e3·31-s + 1.26e3·37-s − 2.47e4·47-s − 1.68e4·49-s − 3.48e4·53-s + 2.48e4·59-s − 3.27e4·64-s − 7.29e4·67-s + 6.62e4·71-s − 5.83e4·80-s + 9.10e4·89-s + 3.13e4·92-s − 1.63e5·97-s − 3.96e3·100-s − 1.80e5·103-s − 1.92e5·113-s + 5.59e4·115-s + ⋯ |
| L(s) = 1 | − 4-s − 1.01·5-s + 16-s + 1.01·20-s − 0.386·23-s + 0.0396·25-s − 1.45·31-s + 0.152·37-s − 1.63·47-s − 49-s − 1.70·53-s + 0.928·59-s − 64-s − 1.98·67-s + 1.56·71-s − 1.01·80-s + 1.21·89-s + 0.386·92-s − 1.76·97-s − 0.0396·100-s − 1.67·103-s − 1.41·113-s + 0.394·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4551026693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4551026693\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + p^{5} T^{2} \) |
| 5 | \( 1 + 57 T + p^{5} T^{2} \) |
| 7 | \( 1 + p^{5} T^{2} \) |
| 13 | \( 1 + p^{5} T^{2} \) |
| 17 | \( 1 + p^{5} T^{2} \) |
| 19 | \( 1 + p^{5} T^{2} \) |
| 23 | \( 1 + 981 T + p^{5} T^{2} \) |
| 29 | \( 1 + p^{5} T^{2} \) |
| 31 | \( 1 + 7775 T + p^{5} T^{2} \) |
| 37 | \( 1 - 1267 T + p^{5} T^{2} \) |
| 41 | \( 1 + p^{5} T^{2} \) |
| 43 | \( 1 + p^{5} T^{2} \) |
| 47 | \( 1 + 24708 T + p^{5} T^{2} \) |
| 53 | \( 1 + 34806 T + p^{5} T^{2} \) |
| 59 | \( 1 - 24825 T + p^{5} T^{2} \) |
| 61 | \( 1 + p^{5} T^{2} \) |
| 67 | \( 1 + 72917 T + p^{5} T^{2} \) |
| 71 | \( 1 - 66273 T + p^{5} T^{2} \) |
| 73 | \( 1 + p^{5} T^{2} \) |
| 79 | \( 1 + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 - 91089 T + p^{5} T^{2} \) |
| 97 | \( 1 + 163183 T + p^{5} T^{2} \) |
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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
−9.160864104662696401489263529000, −8.162086142865290679553310799273, −7.80694677483167011841192298221, −6.69720928274375788433176940973, −5.57662028299858158121151760809, −4.70604443390278060854479623455, −3.91708696298898488508497423673, −3.19240183025675112265815492861, −1.60033307343452145565478844143, −0.29616662609046086355939792127,
0.29616662609046086355939792127, 1.60033307343452145565478844143, 3.19240183025675112265815492861, 3.91708696298898488508497423673, 4.70604443390278060854479623455, 5.57662028299858158121151760809, 6.69720928274375788433176940973, 7.80694677483167011841192298221, 8.162086142865290679553310799273, 9.160864104662696401489263529000
