| L(s) = 1 | + 3-s − 2·7-s + 9-s + 11-s + 2·13-s + 2·19-s − 2·21-s − 5·25-s + 27-s − 10·29-s − 4·31-s + 33-s − 2·37-s + 2·39-s − 2·41-s + 2·43-s + 8·47-s − 3·49-s + 4·53-s + 2·57-s + 12·59-s + 6·61-s − 2·63-s + 2·67-s − 6·73-s − 5·75-s − 2·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.458·19-s − 0.436·21-s − 25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.174·33-s − 0.328·37-s + 0.320·39-s − 0.312·41-s + 0.304·43-s + 1.16·47-s − 3/7·49-s + 0.549·53-s + 0.264·57-s + 1.56·59-s + 0.768·61-s − 0.251·63-s + 0.244·67-s − 0.702·73-s − 0.577·75-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 279312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 279312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 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
−13.03931683422726, −12.69906468763586, −12.06048433982397, −11.56607009166248, −11.24627736748318, −10.61300740708451, −10.12529487015797, −9.695078556110499, −9.275191416565122, −8.871763786581405, −8.468447070860199, −7.691239081767621, −7.509003489542897, −6.901038986664132, −6.469956285578612, −5.743174209245127, −5.582360453107278, −4.857102195257622, −4.032981330261132, −3.685541795992525, −3.476241568049668, −2.626330470062570, −2.118940129700068, −1.556458007616710, −0.7876483513971663, 0,
0.7876483513971663, 1.556458007616710, 2.118940129700068, 2.626330470062570, 3.476241568049668, 3.685541795992525, 4.032981330261132, 4.857102195257622, 5.582360453107278, 5.743174209245127, 6.469956285578612, 6.901038986664132, 7.509003489542897, 7.691239081767621, 8.468447070860199, 8.871763786581405, 9.275191416565122, 9.695078556110499, 10.12529487015797, 10.61300740708451, 11.24627736748318, 11.56607009166248, 12.06048433982397, 12.69906468763586, 13.03931683422726
