| L(s) = 1 | − 3-s − 5-s + 7-s − 2·9-s + 11-s + 15-s − 2·17-s − 2·19-s − 21-s − 7·23-s − 4·25-s + 5·27-s − 10·29-s + 7·31-s − 33-s − 35-s − 9·37-s − 2·41-s − 4·43-s + 2·45-s + 8·47-s + 49-s + 2·51-s + 2·53-s − 55-s + 2·57-s − 15·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s − 2/3·9-s + 0.301·11-s + 0.258·15-s − 0.485·17-s − 0.458·19-s − 0.218·21-s − 1.45·23-s − 4/5·25-s + 0.962·27-s − 1.85·29-s + 1.25·31-s − 0.174·33-s − 0.169·35-s − 1.47·37-s − 0.312·41-s − 0.609·43-s + 0.298·45-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s − 0.134·55-s + 0.264·57-s − 1.95·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−10.40803452533881540695459677647, −9.278084373230974675371114295586, −8.381688983007045538161211097648, −7.60733481301599938371178107590, −6.43901603613253622815630535531, −5.68528498119045167588778097349, −4.59462644263093439550159250376, −3.57850209931108629116064534483, −2.00157538794055782907994458283, 0,
2.00157538794055782907994458283, 3.57850209931108629116064534483, 4.59462644263093439550159250376, 5.68528498119045167588778097349, 6.43901603613253622815630535531, 7.60733481301599938371178107590, 8.381688983007045538161211097648, 9.278084373230974675371114295586, 10.40803452533881540695459677647
