| L(s) = 1 | + 2·3-s + 7-s + 9-s − 4·11-s − 2·13-s + 4·17-s + 2·21-s − 23-s − 5·25-s − 4·27-s − 29-s + 10·31-s − 8·33-s − 2·37-s − 4·39-s − 6·41-s + 4·43-s − 2·47-s + 49-s + 8·51-s + 6·53-s + 10·59-s + 63-s − 4·67-s − 2·69-s − 8·71-s + 14·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.970·17-s + 0.436·21-s − 0.208·23-s − 25-s − 0.769·27-s − 0.185·29-s + 1.79·31-s − 1.39·33-s − 0.328·37-s − 0.640·39-s − 0.937·41-s + 0.609·43-s − 0.291·47-s + 1/7·49-s + 1.12·51-s + 0.824·53-s + 1.30·59-s + 0.125·63-s − 0.488·67-s − 0.240·69-s − 0.949·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.27565405542675, −13.85336031827670, −13.42732477299978, −13.03291659963596, −12.30977178628242, −11.84646298459623, −11.46772231726337, −10.60677616824111, −10.18518988359620, −9.807766695505411, −9.292167361145490, −8.459238189928009, −8.313650898444373, −7.716588877406217, −7.445818088850630, −6.700548906438415, −5.876998350957910, −5.432052619517101, −4.862136324379416, −4.187027218620793, −3.524036165364467, −2.975182072262247, −2.396472541862337, −1.968111706340382, −0.9957895646584747, 0,
0.9957895646584747, 1.968111706340382, 2.396472541862337, 2.975182072262247, 3.524036165364467, 4.187027218620793, 4.862136324379416, 5.432052619517101, 5.876998350957910, 6.700548906438415, 7.445818088850630, 7.716588877406217, 8.313650898444373, 8.459238189928009, 9.292167361145490, 9.807766695505411, 10.18518988359620, 10.60677616824111, 11.46772231726337, 11.84646298459623, 12.30977178628242, 13.03291659963596, 13.42732477299978, 13.85336031827670, 14.27565405542675
