| L(s) = 1 | − 3-s − 5-s + 9-s − 3·11-s + 4·13-s + 15-s + 4·19-s − 8·23-s − 4·25-s − 27-s − 3·29-s + 5·31-s + 3·33-s + 8·37-s − 4·39-s + 8·41-s − 6·43-s − 45-s − 10·47-s + 9·53-s + 3·55-s − 4·57-s + 5·59-s − 10·61-s − 4·65-s − 6·67-s + 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.904·11-s + 1.10·13-s + 0.258·15-s + 0.917·19-s − 1.66·23-s − 4/5·25-s − 0.192·27-s − 0.557·29-s + 0.898·31-s + 0.522·33-s + 1.31·37-s − 0.640·39-s + 1.24·41-s − 0.914·43-s − 0.149·45-s − 1.45·47-s + 1.23·53-s + 0.404·55-s − 0.529·57-s + 0.650·59-s − 1.28·61-s − 0.496·65-s − 0.733·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
| 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
−8.334890438041431228113248694644, −7.921396703956260164318906156922, −7.11523404301419379499703193329, −6.00767996510914171325338296590, −5.68534372425012231445925199397, −4.52232060046538851571999455882, −3.83219049015790168921238875875, −2.74547262646093177949303742151, −1.40932437509895629798638039361, 0,
1.40932437509895629798638039361, 2.74547262646093177949303742151, 3.83219049015790168921238875875, 4.52232060046538851571999455882, 5.68534372425012231445925199397, 6.00767996510914171325338296590, 7.11523404301419379499703193329, 7.921396703956260164318906156922, 8.334890438041431228113248694644
