| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s − 2·7-s + 9-s − 2·12-s + 3·13-s − 4·14-s − 4·16-s − 6·17-s + 2·18-s + 2·19-s + 2·21-s − 4·23-s − 5·25-s + 6·26-s − 27-s − 4·28-s + 2·29-s − 8·31-s − 8·32-s − 12·34-s + 2·36-s − 3·37-s + 4·38-s − 3·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 0.755·7-s + 1/3·9-s − 0.577·12-s + 0.832·13-s − 1.06·14-s − 16-s − 1.45·17-s + 0.471·18-s + 0.458·19-s + 0.436·21-s − 0.834·23-s − 25-s + 1.17·26-s − 0.192·27-s − 0.755·28-s + 0.371·29-s − 1.43·31-s − 1.41·32-s − 2.05·34-s + 1/3·36-s − 0.493·37-s + 0.648·38-s − 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 677 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
| 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
−8.931019517839741083537471197119, −7.68141092737707684390382126625, −6.76058779586649748576118406887, −6.09637543753631838705696689257, −5.66953150384102211634173759016, −4.57356449426204066571909794624, −3.96151466586683929846590092808, −3.12341686864696477942526249639, −1.93620944174678157553749068566, 0,
1.93620944174678157553749068566, 3.12341686864696477942526249639, 3.96151466586683929846590092808, 4.57356449426204066571909794624, 5.66953150384102211634173759016, 6.09637543753631838705696689257, 6.76058779586649748576118406887, 7.68141092737707684390382126625, 8.931019517839741083537471197119
