| L(s) = 1 | + 3·3-s + 4·7-s + 6·9-s − 3·11-s + 6·13-s − 5·17-s + 19-s + 12·21-s + 23-s + 9·27-s − 8·29-s + 8·31-s − 9·33-s − 2·37-s + 18·39-s − 7·41-s + 4·43-s + 10·47-s + 9·49-s − 15·51-s + 12·53-s + 3·57-s − 4·59-s − 8·61-s + 24·63-s + 3·67-s + 3·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.51·7-s + 2·9-s − 0.904·11-s + 1.66·13-s − 1.21·17-s + 0.229·19-s + 2.61·21-s + 0.208·23-s + 1.73·27-s − 1.48·29-s + 1.43·31-s − 1.56·33-s − 0.328·37-s + 2.88·39-s − 1.09·41-s + 0.609·43-s + 1.45·47-s + 9/7·49-s − 2.10·51-s + 1.64·53-s + 0.397·57-s − 0.520·59-s − 1.02·61-s + 3.02·63-s + 0.366·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.377205349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.377205349\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−7.83946552690032228270618185469, −7.42585454809721811189304598057, −6.51707370121996074186479265497, −5.55206439653174992373942392433, −4.74728800647633660983169779413, −4.07665131216204267933897727968, −3.45559536366122160998013797471, −2.47940396947319998410163922686, −1.96645933355497173012948667178, −1.10909854550830689275189128881,
1.10909854550830689275189128881, 1.96645933355497173012948667178, 2.47940396947319998410163922686, 3.45559536366122160998013797471, 4.07665131216204267933897727968, 4.74728800647633660983169779413, 5.55206439653174992373942392433, 6.51707370121996074186479265497, 7.42585454809721811189304598057, 7.83946552690032228270618185469
