L(s) = 1 | + 3-s + 5-s + 3·7-s + 9-s − 4·11-s + 2·13-s + 15-s − 6·17-s − 2·19-s + 3·21-s − 6·23-s − 4·25-s + 27-s + 3·31-s − 4·33-s + 3·35-s + 7·37-s + 2·39-s − 6·41-s − 12·43-s + 45-s − 3·47-s + 2·49-s − 6·51-s − 9·53-s − 4·55-s − 2·57-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.458·19-s + 0.654·21-s − 1.25·23-s − 4/5·25-s + 0.192·27-s + 0.538·31-s − 0.696·33-s + 0.507·35-s + 1.15·37-s + 0.320·39-s − 0.937·41-s − 1.82·43-s + 0.149·45-s − 0.437·47-s + 2/7·49-s − 0.840·51-s − 1.23·53-s − 0.539·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.84656839393951240373732304785, −6.73540444900842065319873945894, −6.18529542637951848860977916066, −5.27643168238538488129750628137, −4.65525584342180932039932063144, −4.00865590761489484185968329770, −2.94592462754609146169906368995, −2.09343796088768158702854174686, −1.65125777954587527391259496118, 0,
1.65125777954587527391259496118, 2.09343796088768158702854174686, 2.94592462754609146169906368995, 4.00865590761489484185968329770, 4.65525584342180932039932063144, 5.27643168238538488129750628137, 6.18529542637951848860977916066, 6.73540444900842065319873945894, 7.84656839393951240373732304785