L(s) = 1 | − 3-s − 5-s − 2·7-s + 9-s + 6·11-s − 4·13-s + 15-s − 6·17-s + 4·19-s + 2·21-s + 25-s − 27-s + 29-s − 2·31-s − 6·33-s + 2·35-s + 2·37-s + 4·39-s + 4·43-s − 45-s − 3·49-s + 6·51-s + 6·53-s − 6·55-s − 4·57-s + 12·59-s − 4·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s + 1.80·11-s − 1.10·13-s + 0.258·15-s − 1.45·17-s + 0.917·19-s + 0.436·21-s + 1/5·25-s − 0.192·27-s + 0.185·29-s − 0.359·31-s − 1.04·33-s + 0.338·35-s + 0.328·37-s + 0.640·39-s + 0.609·43-s − 0.149·45-s − 3/7·49-s + 0.840·51-s + 0.824·53-s − 0.809·55-s − 0.529·57-s + 1.56·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.21877987157052444188125769416, −6.98712750804345428426098298380, −6.31455599927065416000467941347, −5.56490728425822153742119965938, −4.59329019494136964032290932283, −4.10719299524357991189994101015, −3.27791750896394206631893451293, −2.26605445307048913791463437356, −1.10245495945180345525249057119, 0,
1.10245495945180345525249057119, 2.26605445307048913791463437356, 3.27791750896394206631893451293, 4.10719299524357991189994101015, 4.59329019494136964032290932283, 5.56490728425822153742119965938, 6.31455599927065416000467941347, 6.98712750804345428426098298380, 7.21877987157052444188125769416