| 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
| \(L(1)\) |
\(\approx\) |
\(1.173873920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.173873920\) |
| \(L(\frac{3}{2})\) |
|
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
−9.523220157361629673806012120499, −8.656210371167276556599811060874, −7.65435242334397075510757498968, −6.92639756693930427749807560382, −6.09693943919241168618876642066, −5.25764030260127216381141022318, −4.67194703335490856544942450868, −3.57697646553513772381900969499, −2.09524480724426133048492181096, −0.820603880234232144260936979376,
0.820603880234232144260936979376, 2.09524480724426133048492181096, 3.57697646553513772381900969499, 4.67194703335490856544942450868, 5.25764030260127216381141022318, 6.09693943919241168618876642066, 6.92639756693930427749807560382, 7.65435242334397075510757498968, 8.656210371167276556599811060874, 9.523220157361629673806012120499
