L(s) = 1 | + 2·3-s − 2·7-s + 9-s + 4·11-s + 4·13-s + 4·19-s − 4·21-s + 2·23-s − 4·27-s − 2·29-s + 8·33-s + 4·37-s + 8·39-s + 2·41-s − 6·43-s + 6·47-s − 3·49-s − 4·53-s + 8·57-s + 12·59-s + 10·61-s − 2·63-s + 14·67-s + 4·69-s + 8·71-s − 8·73-s − 8·77-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s + 0.917·19-s − 0.872·21-s + 0.417·23-s − 0.769·27-s − 0.371·29-s + 1.39·33-s + 0.657·37-s + 1.28·39-s + 0.312·41-s − 0.914·43-s + 0.875·47-s − 3/7·49-s − 0.549·53-s + 1.05·57-s + 1.56·59-s + 1.28·61-s − 0.251·63-s + 1.71·67-s + 0.481·69-s + 0.949·71-s − 0.936·73-s − 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.624861882\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.624861882\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−9.430807639724219831428449893144, −8.653058555254410798042808386384, −8.030431749465228335381340080762, −6.99503062258751160883576746513, −6.34228894461127709353829638947, −5.35262082597715768581833218902, −3.87023051487882198452611809156, −3.53327560811031697417327084464, −2.48592289355078388615496718721, −1.17328522857385290428632799862,
1.17328522857385290428632799862, 2.48592289355078388615496718721, 3.53327560811031697417327084464, 3.87023051487882198452611809156, 5.35262082597715768581833218902, 6.34228894461127709353829638947, 6.99503062258751160883576746513, 8.030431749465228335381340080762, 8.653058555254410798042808386384, 9.430807639724219831428449893144