| L(s) = 1 | + 3-s + 2·5-s + 4·7-s + 9-s + 4·11-s − 2·13-s + 2·15-s − 2·17-s + 4·21-s − 23-s − 25-s + 27-s − 2·29-s + 4·33-s + 8·35-s − 10·37-s − 2·39-s − 6·41-s − 8·43-s + 2·45-s + 8·47-s + 9·49-s − 2·51-s − 6·53-s + 8·55-s + 4·59-s + 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.516·15-s − 0.485·17-s + 0.872·21-s − 0.208·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 0.696·33-s + 1.35·35-s − 1.64·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.298·45-s + 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.824·53-s + 1.07·55-s + 0.520·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.765224801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.765224801\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.785842552178354779577446812151, −8.932060008579692298515103315193, −8.395671715216651537140155732006, −7.39054854691244144388329080841, −6.59119424439838566457949725248, −5.46798261913793907985091608260, −4.67361052054189892229166057577, −3.66546032760679994855744751045, −2.16190010413024304208365545860, −1.55057711786882841910328318755,
1.55057711786882841910328318755, 2.16190010413024304208365545860, 3.66546032760679994855744751045, 4.67361052054189892229166057577, 5.46798261913793907985091608260, 6.59119424439838566457949725248, 7.39054854691244144388329080841, 8.395671715216651537140155732006, 8.932060008579692298515103315193, 9.785842552178354779577446812151
