| L(s) = 1 | − 3-s − 5-s − 2·9-s − 11-s + 4·13-s + 15-s − 6·17-s − 2·19-s − 9·23-s + 25-s + 5·27-s + 9·29-s + 4·31-s + 33-s + 2·37-s − 4·39-s − 3·41-s + 11·43-s + 2·45-s + 12·47-s + 6·51-s + 55-s + 2·57-s + 7·61-s − 4·65-s + 5·67-s + 9·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.301·11-s + 1.10·13-s + 0.258·15-s − 1.45·17-s − 0.458·19-s − 1.87·23-s + 1/5·25-s + 0.962·27-s + 1.67·29-s + 0.718·31-s + 0.174·33-s + 0.328·37-s − 0.640·39-s − 0.468·41-s + 1.67·43-s + 0.298·45-s + 1.75·47-s + 0.840·51-s + 0.134·55-s + 0.264·57-s + 0.896·61-s − 0.496·65-s + 0.610·67-s + 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−16.90577569627787, −15.91994022999472, −15.86383408531792, −15.43884502959586, −14.28067856592035, −14.10812260169424, −13.37988290600961, −12.73525563828917, −12.07856306973311, −11.58898867727913, −11.09539850234481, −10.48791950692025, −10.04961982796493, −8.857317481326777, −8.629071153085343, −8.034754176834296, −7.196999701150778, −6.258146874829633, −6.176027443709012, −5.295808297087206, −4.301033127137961, −4.087379318388491, −2.878172435199233, −2.256504095732686, −0.9646738707022310, 0,
0.9646738707022310, 2.256504095732686, 2.878172435199233, 4.087379318388491, 4.301033127137961, 5.295808297087206, 6.176027443709012, 6.258146874829633, 7.196999701150778, 8.034754176834296, 8.629071153085343, 8.857317481326777, 10.04961982796493, 10.48791950692025, 11.09539850234481, 11.58898867727913, 12.07856306973311, 12.73525563828917, 13.37988290600961, 14.10812260169424, 14.28067856592035, 15.43884502959586, 15.86383408531792, 15.91994022999472, 16.90577569627787
