| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 2·11-s − 4·13-s + 2·14-s + 16-s − 2·17-s − 19-s − 2·22-s + 4·23-s − 4·26-s + 2·28-s − 8·31-s + 32-s − 2·34-s − 8·37-s − 38-s + 8·41-s + 6·43-s − 2·44-s + 4·46-s − 12·47-s − 3·49-s − 4·52-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 0.603·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s − 0.229·19-s − 0.426·22-s + 0.834·23-s − 0.784·26-s + 0.377·28-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 1.31·37-s − 0.162·38-s + 1.24·41-s + 0.914·43-s − 0.301·44-s + 0.589·46-s − 1.75·47-s − 3/7·49-s − 0.554·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
| 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
−7.43497398753394446325207128900, −6.77314924840295592221284830831, −5.93309863544358775579937097129, −5.06486408141361635860126041329, −4.87850886269875884523769208147, −3.97790187284038039780780400231, −3.05946013742975761460459534844, −2.31113594193571247930445376114, −1.53081178679857996612768776453, 0,
1.53081178679857996612768776453, 2.31113594193571247930445376114, 3.05946013742975761460459534844, 3.97790187284038039780780400231, 4.87850886269875884523769208147, 5.06486408141361635860126041329, 5.93309863544358775579937097129, 6.77314924840295592221284830831, 7.43497398753394446325207128900
