| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s + 11-s + 4·13-s + 5·17-s + 19-s − 6·21-s − 2·23-s − 9·27-s − 8·29-s + 10·31-s − 3·33-s − 6·37-s − 12·39-s − 3·41-s + 4·43-s + 4·47-s − 3·49-s − 15·51-s + 6·53-s − 3·57-s + 8·59-s + 10·61-s + 12·63-s − 67-s + 6·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s + 0.301·11-s + 1.10·13-s + 1.21·17-s + 0.229·19-s − 1.30·21-s − 0.417·23-s − 1.73·27-s − 1.48·29-s + 1.79·31-s − 0.522·33-s − 0.986·37-s − 1.92·39-s − 0.468·41-s + 0.609·43-s + 0.583·47-s − 3/7·49-s − 2.10·51-s + 0.824·53-s − 0.397·57-s + 1.04·59-s + 1.28·61-s + 1.51·63-s − 0.122·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8086102703\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8086102703\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 13 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 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
−12.07137470450026695132786865615, −11.57330873285149056321996472414, −10.73352697394264976620509349656, −9.870419190423355026152723289977, −8.343465400986863609159475132154, −7.12229165851766752840688929571, −5.98579749846617233669316674897, −5.26260943901165948452216050736, −3.98257276053604497857370793582, −1.24881502125002515972535331485,
1.24881502125002515972535331485, 3.98257276053604497857370793582, 5.26260943901165948452216050736, 5.98579749846617233669316674897, 7.12229165851766752840688929571, 8.343465400986863609159475132154, 9.870419190423355026152723289977, 10.73352697394264976620509349656, 11.57330873285149056321996472414, 12.07137470450026695132786865615
