| L(s) = 1 | + 3·3-s + 5-s + 6·9-s + 2·11-s + 3·15-s + 4·17-s − 6·19-s − 3·23-s + 25-s + 9·27-s + 9·29-s − 4·31-s + 6·33-s − 4·37-s + 7·41-s + 5·43-s + 6·45-s + 8·47-s + 12·51-s − 2·53-s + 2·55-s − 18·57-s + 10·59-s − 61-s + 9·67-s − 9·69-s − 2·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s + 0.603·11-s + 0.774·15-s + 0.970·17-s − 1.37·19-s − 0.625·23-s + 1/5·25-s + 1.73·27-s + 1.67·29-s − 0.718·31-s + 1.04·33-s − 0.657·37-s + 1.09·41-s + 0.762·43-s + 0.894·45-s + 1.16·47-s + 1.68·51-s − 0.274·53-s + 0.269·55-s − 2.38·57-s + 1.30·59-s − 0.128·61-s + 1.09·67-s − 1.08·69-s − 0.237·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.344137996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.344137996\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 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
−8.539689340899603437218648934153, −7.937699962953917311704821184553, −7.14491549470949409213015301847, −6.43300899613947666621042164956, −5.50674070832460457445830359535, −4.29768021898691769762074626596, −3.83639934130619610135409165779, −2.82544846676668415349152148721, −2.19810415966871176781863442736, −1.22003126041481978230606908800,
1.22003126041481978230606908800, 2.19810415966871176781863442736, 2.82544846676668415349152148721, 3.83639934130619610135409165779, 4.29768021898691769762074626596, 5.50674070832460457445830359535, 6.43300899613947666621042164956, 7.14491549470949409213015301847, 7.937699962953917311704821184553, 8.539689340899603437218648934153
