| L(s) = 1 | + 3-s − 5-s + 7-s − 2·9-s + 11-s + 2·13-s − 15-s − 5·17-s − 7·19-s + 21-s + 6·23-s + 25-s − 5·27-s + 29-s + 5·31-s + 33-s − 35-s − 11·37-s + 2·39-s + 2·41-s + 4·43-s + 2·45-s − 6·47-s − 6·49-s − 5·51-s + 53-s − 55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s − 1.21·17-s − 1.60·19-s + 0.218·21-s + 1.25·23-s + 1/5·25-s − 0.962·27-s + 0.185·29-s + 0.898·31-s + 0.174·33-s − 0.169·35-s − 1.80·37-s + 0.320·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s − 0.875·47-s − 6/7·49-s − 0.700·51-s + 0.137·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 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
−8.418111800052195344722671462773, −7.60663664842651683300516800313, −6.66360260277600915618739814781, −6.15369215637965608236546056019, −4.96006477680199860624059474360, −4.33038809622237587996842730816, −3.42005218564476194074585728841, −2.59825552753108262616751280567, −1.59062078455961220686220511077, 0,
1.59062078455961220686220511077, 2.59825552753108262616751280567, 3.42005218564476194074585728841, 4.33038809622237587996842730816, 4.96006477680199860624059474360, 6.15369215637965608236546056019, 6.66360260277600915618739814781, 7.60663664842651683300516800313, 8.418111800052195344722671462773
