L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s + 2·11-s − 12-s − 6·13-s − 14-s + 16-s − 4·17-s − 18-s − 6·19-s − 21-s − 2·22-s + 8·23-s + 24-s + 6·26-s − 27-s + 28-s + 6·29-s − 2·31-s − 32-s − 2·33-s + 4·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s − 1.37·19-s − 0.218·21-s − 0.426·22-s + 1.66·23-s + 0.204·24-s + 1.17·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.359·31-s − 0.176·32-s − 0.348·33-s + 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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
−9.426178265642867577600024743995, −8.825803983912225731574364653776, −7.83723813654224614629680182817, −6.89570877143456989605578258768, −6.43868687161705628380646564779, −5.06819747119938962105631935673, −4.44322897549371915935408752381, −2.81944329983774384015026078695, −1.64378675367087627115681173475, 0,
1.64378675367087627115681173475, 2.81944329983774384015026078695, 4.44322897549371915935408752381, 5.06819747119938962105631935673, 6.43868687161705628380646564779, 6.89570877143456989605578258768, 7.83723813654224614629680182817, 8.825803983912225731574364653776, 9.426178265642867577600024743995