| L(s) = 1 | − 3-s + 3·7-s + 9-s − 3·11-s + 13-s + 17-s − 6·19-s − 3·21-s + 5·23-s − 27-s − 6·29-s + 2·31-s + 3·33-s − 7·37-s − 39-s + 3·41-s + 8·43-s + 2·47-s + 2·49-s − 51-s + 53-s + 6·57-s − 15·61-s + 3·63-s − 12·67-s − 5·69-s + 5·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.242·17-s − 1.37·19-s − 0.654·21-s + 1.04·23-s − 0.192·27-s − 1.11·29-s + 0.359·31-s + 0.522·33-s − 1.15·37-s − 0.160·39-s + 0.468·41-s + 1.21·43-s + 0.291·47-s + 2/7·49-s − 0.140·51-s + 0.137·53-s + 0.794·57-s − 1.92·61-s + 0.377·63-s − 1.46·67-s − 0.601·69-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−7.50511904331732765022984398424, −6.90113601794186469340473580057, −5.92828368902470378791516516037, −5.46890964498597889992225469236, −4.68459107374662844322886347686, −4.19272152033327055800018711711, −3.06125026840720023214306365744, −2.10590691770361227509380017663, −1.28070790879119080956651091892, 0,
1.28070790879119080956651091892, 2.10590691770361227509380017663, 3.06125026840720023214306365744, 4.19272152033327055800018711711, 4.68459107374662844322886347686, 5.46890964498597889992225469236, 5.92828368902470378791516516037, 6.90113601794186469340473580057, 7.50511904331732765022984398424
