| L(s) = 1 | + 18·7-s + 34·11-s − 12·13-s + 102·17-s + 164·19-s − 48·23-s + 146·29-s + 100·31-s − 328·37-s − 288·41-s − 120·43-s − 16·47-s − 19·49-s + 126·53-s + 642·59-s + 602·61-s − 436·67-s + 652·71-s − 1.06e3·73-s + 612·77-s + 388·79-s + 444·83-s − 820·89-s − 216·91-s + 766·97-s − 798·101-s + 402·103-s + ⋯ |
| L(s) = 1 | + 0.971·7-s + 0.931·11-s − 0.256·13-s + 1.45·17-s + 1.98·19-s − 0.435·23-s + 0.934·29-s + 0.579·31-s − 1.45·37-s − 1.09·41-s − 0.425·43-s − 0.0496·47-s − 0.0553·49-s + 0.326·53-s + 1.41·59-s + 1.26·61-s − 0.795·67-s + 1.08·71-s − 1.70·73-s + 0.905·77-s + 0.552·79-s + 0.587·83-s − 0.976·89-s − 0.248·91-s + 0.801·97-s − 0.786·101-s + 0.384·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.139191607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.139191607\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 18 T + p^{3} T^{2} \) |
| 11 | \( 1 - 34 T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 19 | \( 1 - 164 T + p^{3} T^{2} \) |
| 23 | \( 1 + 48 T + p^{3} T^{2} \) |
| 29 | \( 1 - 146 T + p^{3} T^{2} \) |
| 31 | \( 1 - 100 T + p^{3} T^{2} \) |
| 37 | \( 1 + 328 T + p^{3} T^{2} \) |
| 41 | \( 1 + 288 T + p^{3} T^{2} \) |
| 43 | \( 1 + 120 T + p^{3} T^{2} \) |
| 47 | \( 1 + 16 T + p^{3} T^{2} \) |
| 53 | \( 1 - 126 T + p^{3} T^{2} \) |
| 59 | \( 1 - 642 T + p^{3} T^{2} \) |
| 61 | \( 1 - 602 T + p^{3} T^{2} \) |
| 67 | \( 1 + 436 T + p^{3} T^{2} \) |
| 71 | \( 1 - 652 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1062 T + p^{3} T^{2} \) |
| 79 | \( 1 - 388 T + p^{3} T^{2} \) |
| 83 | \( 1 - 444 T + p^{3} T^{2} \) |
| 89 | \( 1 + 820 T + p^{3} T^{2} \) |
| 97 | \( 1 - 766 T + p^{3} T^{2} \) |
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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
−8.815480638264399447975459988290, −8.120799627742510632978125089546, −7.40346131871326765472122093729, −6.62587344281596372153535650226, −5.46122191479674465776888336357, −5.01512986003752922979247077591, −3.85414477658000715513048361940, −3.04155709110035465919603459063, −1.65431251722322081903760929178, −0.921920790989417790517372818273,
0.921920790989417790517372818273, 1.65431251722322081903760929178, 3.04155709110035465919603459063, 3.85414477658000715513048361940, 5.01512986003752922979247077591, 5.46122191479674465776888336357, 6.62587344281596372153535650226, 7.40346131871326765472122093729, 8.120799627742510632978125089546, 8.815480638264399447975459988290
