| L(s) = 1 | + 2·3-s + 4·7-s + 9-s + 6·11-s − 2·13-s + 4·19-s + 8·21-s − 4·27-s − 4·31-s + 12·33-s − 4·37-s − 4·39-s − 6·41-s + 8·43-s + 9·49-s + 6·53-s + 8·57-s + 4·61-s + 4·63-s + 8·67-s + 2·73-s + 24·77-s + 8·79-s − 11·81-s − 6·89-s − 8·91-s − 8·93-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s + 1.80·11-s − 0.554·13-s + 0.917·19-s + 1.74·21-s − 0.769·27-s − 0.718·31-s + 2.08·33-s − 0.657·37-s − 0.640·39-s − 0.937·41-s + 1.21·43-s + 9/7·49-s + 0.824·53-s + 1.05·57-s + 0.512·61-s + 0.503·63-s + 0.977·67-s + 0.234·73-s + 2.73·77-s + 0.900·79-s − 1.22·81-s − 0.635·89-s − 0.838·91-s − 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.470341578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.470341578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.96582087213675, −13.28027949360342, −12.63750040451009, −11.97022486813255, −11.74102261104532, −11.29762498234281, −10.75977634309922, −10.04192420637618, −9.520426933616849, −9.093690524406129, −8.677039262265176, −8.305512966092808, −7.609551022605158, −7.362936254504092, −6.774324012176943, −6.062099599822776, −5.355188945904279, −4.958535434917611, −4.229472464003836, −3.764942303022496, −3.326112791398150, −2.462707098656007, −1.946222684807476, −1.448378547336679, −0.7422829552441614,
0.7422829552441614, 1.448378547336679, 1.946222684807476, 2.462707098656007, 3.326112791398150, 3.764942303022496, 4.229472464003836, 4.958535434917611, 5.355188945904279, 6.062099599822776, 6.774324012176943, 7.362936254504092, 7.609551022605158, 8.305512966092808, 8.677039262265176, 9.093690524406129, 9.520426933616849, 10.04192420637618, 10.75977634309922, 11.29762498234281, 11.74102261104532, 11.97022486813255, 12.63750040451009, 13.28027949360342, 13.96582087213675
