| L(s) = 1 | − 3-s − 2·5-s + 9-s + 11-s + 13-s + 2·15-s − 6·17-s + 4·19-s + 8·23-s − 25-s − 27-s − 10·29-s − 33-s + 6·37-s − 39-s + 10·41-s − 4·43-s − 2·45-s − 8·47-s − 7·49-s + 6·51-s − 10·53-s − 2·55-s − 4·57-s + 12·59-s + 14·61-s − 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 0.174·33-s + 0.986·37-s − 0.160·39-s + 1.56·41-s − 0.609·43-s − 0.298·45-s − 1.16·47-s − 49-s + 0.840·51-s − 1.37·53-s − 0.269·55-s − 0.529·57-s + 1.56·59-s + 1.79·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.049022738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.049022738\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−17.38335752989767, −16.50022924791857, −16.01736062161417, −15.64866410906929, −14.79062662841783, −14.57634974427321, −13.34717090749148, −13.08886336965384, −12.49537775797773, −11.40624082888509, −11.31781876523701, −11.03715970424672, −9.744686640339811, −9.434145498060229, −8.555371669739059, −7.922331499607123, −7.106301351258225, −6.762604593997035, −5.807355979841270, −5.120839812501957, −4.330467788472542, −3.746505927835664, −2.835955623103920, −1.653231786711949, −0.5483736276693433,
0.5483736276693433, 1.653231786711949, 2.835955623103920, 3.746505927835664, 4.330467788472542, 5.120839812501957, 5.807355979841270, 6.762604593997035, 7.106301351258225, 7.922331499607123, 8.555371669739059, 9.434145498060229, 9.744686640339811, 11.03715970424672, 11.31781876523701, 11.40624082888509, 12.49537775797773, 13.08886336965384, 13.34717090749148, 14.57634974427321, 14.79062662841783, 15.64866410906929, 16.01736062161417, 16.50022924791857, 17.38335752989767
