| L(s) = 1 | − 5-s − 7-s − 3·11-s − 13-s + 3·17-s + 2·19-s + 6·23-s + 25-s + 29-s − 4·31-s + 35-s − 4·37-s + 6·41-s − 4·43-s + 3·47-s − 6·49-s − 6·53-s + 3·55-s + 8·61-s + 65-s + 5·67-s + 6·71-s − 10·73-s + 3·77-s + 8·79-s + 6·83-s − 3·85-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.904·11-s − 0.277·13-s + 0.727·17-s + 0.458·19-s + 1.25·23-s + 1/5·25-s + 0.185·29-s − 0.718·31-s + 0.169·35-s − 0.657·37-s + 0.937·41-s − 0.609·43-s + 0.437·47-s − 6/7·49-s − 0.824·53-s + 0.404·55-s + 1.02·61-s + 0.124·65-s + 0.610·67-s + 0.712·71-s − 1.17·73-s + 0.341·77-s + 0.900·79-s + 0.658·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.84810196803628289805734009232, −7.17200244107852945270928514398, −6.55871663938078909837613964064, −5.42887452661776211362861662810, −5.12592807923731182988104321368, −4.04737359121481937461253099566, −3.22096091049596195828610343470, −2.58119535601956010950816443634, −1.24733217530574868050256113811, 0,
1.24733217530574868050256113811, 2.58119535601956010950816443634, 3.22096091049596195828610343470, 4.04737359121481937461253099566, 5.12592807923731182988104321368, 5.42887452661776211362861662810, 6.55871663938078909837613964064, 7.17200244107852945270928514398, 7.84810196803628289805734009232
