| L(s) = 1 | − 3-s + 5-s − 3·7-s + 9-s + 3·11-s − 6·13-s − 15-s + 3·17-s + 19-s + 3·21-s − 4·23-s − 4·25-s − 27-s − 10·29-s − 2·31-s − 3·33-s − 3·35-s + 8·37-s + 6·39-s − 8·41-s + 43-s + 45-s − 3·47-s + 2·49-s − 3·51-s − 6·53-s + 3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 1.66·13-s − 0.258·15-s + 0.727·17-s + 0.229·19-s + 0.654·21-s − 0.834·23-s − 4/5·25-s − 0.192·27-s − 1.85·29-s − 0.359·31-s − 0.522·33-s − 0.507·35-s + 1.31·37-s + 0.960·39-s − 1.24·41-s + 0.152·43-s + 0.149·45-s − 0.437·47-s + 2/7·49-s − 0.420·51-s − 0.824·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.721879645093847090940062433997, −9.256845722173249154900109293870, −7.75525037135579232003734806978, −7.04173846168624158743830695656, −6.11989947342759976206118717321, −5.49960710074578838011452836914, −4.29287803140015808576580756413, −3.22481329079945918066760847835, −1.85542976329431191785537331479, 0,
1.85542976329431191785537331479, 3.22481329079945918066760847835, 4.29287803140015808576580756413, 5.49960710074578838011452836914, 6.11989947342759976206118717321, 7.04173846168624158743830695656, 7.75525037135579232003734806978, 9.256845722173249154900109293870, 9.721879645093847090940062433997
