| L(s) = 1 | + 3·3-s − 19·5-s − 9·7-s + 9·9-s + 13·11-s + 38·13-s − 57·15-s + 99·17-s + 19·19-s − 27·21-s − 68·23-s + 236·25-s + 27·27-s + 130·29-s − 262·31-s + 39·33-s + 171·35-s − 296·37-s + 114·39-s − 8·41-s − 73·43-s − 171·45-s + 271·47-s − 262·49-s + 297·51-s − 502·53-s − 247·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.69·5-s − 0.485·7-s + 1/3·9-s + 0.356·11-s + 0.810·13-s − 0.981·15-s + 1.41·17-s + 0.229·19-s − 0.280·21-s − 0.616·23-s + 1.88·25-s + 0.192·27-s + 0.832·29-s − 1.51·31-s + 0.205·33-s + 0.825·35-s − 1.31·37-s + 0.468·39-s − 0.0304·41-s − 0.258·43-s − 0.566·45-s + 0.841·47-s − 0.763·49-s + 0.815·51-s − 1.30·53-s − 0.605·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 19 | \( 1 - p T \) |
| good | 5 | \( 1 + 19 T + p^{3} T^{2} \) |
| 7 | \( 1 + 9 T + p^{3} T^{2} \) |
| 11 | \( 1 - 13 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 99 T + p^{3} T^{2} \) |
| 23 | \( 1 + 68 T + p^{3} T^{2} \) |
| 29 | \( 1 - 130 T + p^{3} T^{2} \) |
| 31 | \( 1 + 262 T + p^{3} T^{2} \) |
| 37 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 8 T + p^{3} T^{2} \) |
| 43 | \( 1 + 73 T + p^{3} T^{2} \) |
| 47 | \( 1 - 271 T + p^{3} T^{2} \) |
| 53 | \( 1 + 502 T + p^{3} T^{2} \) |
| 59 | \( 1 + 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 587 T + p^{3} T^{2} \) |
| 67 | \( 1 + 684 T + p^{3} T^{2} \) |
| 71 | \( 1 + 992 T + p^{3} T^{2} \) |
| 73 | \( 1 + 507 T + p^{3} T^{2} \) |
| 79 | \( 1 + 980 T + p^{3} T^{2} \) |
| 83 | \( 1 - 492 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1046 T + p^{3} T^{2} \) |
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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
−9.094656930672107790275146799580, −8.379868905258049372850795362062, −7.66694394908545797543617202407, −7.01206916618348401808848760469, −5.85545675702482302374071173676, −4.54380280737625464994524070433, −3.57818976021714615311909657834, −3.21626724549331809304706092959, −1.36287741121121468682268487194, 0,
1.36287741121121468682268487194, 3.21626724549331809304706092959, 3.57818976021714615311909657834, 4.54380280737625464994524070433, 5.85545675702482302374071173676, 7.01206916618348401808848760469, 7.66694394908545797543617202407, 8.379868905258049372850795362062, 9.094656930672107790275146799580
