| L(s) = 1 | − 2-s + 4-s − 6·7-s − 8-s + 6·14-s + 16-s + 2·25-s − 6·28-s + 4·29-s − 32-s − 8·41-s − 12·43-s + 17·49-s − 2·50-s − 20·53-s + 6·56-s − 4·58-s + 16·59-s + 64-s + 8·71-s + 22·73-s + 8·82-s + 12·86-s − 17·98-s + 2·100-s + 20·106-s − 16·107-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 2.26·7-s − 0.353·8-s + 1.60·14-s + 1/4·16-s + 2/5·25-s − 1.13·28-s + 0.742·29-s − 0.176·32-s − 1.24·41-s − 1.82·43-s + 17/7·49-s − 0.282·50-s − 2.74·53-s + 0.801·56-s − 0.525·58-s + 2.08·59-s + 1/8·64-s + 0.949·71-s + 2.57·73-s + 0.883·82-s + 1.29·86-s − 1.71·98-s + 1/5·100-s + 1.94·106-s − 1.54·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5538772906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5538772906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 + T \) |
| 3 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 75 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.457880985479899014756103944006, −8.276581415230693607756604397446, −7.73015386086090233925026670194, −6.91964822574493532226298466197, −6.78682437386849717586583618209, −6.41201118200370857374720017569, −6.10107696557531263189876906822, −5.16777488930779092557932302788, −5.01640021464643343700684606261, −3.89908480426948392711504438086, −3.54450326690758281448216229050, −3.01804384251345169358926901809, −2.52646450231481981058882929207, −1.57264325838400092847327723551, −0.43973154708817291664431241670,
0.43973154708817291664431241670, 1.57264325838400092847327723551, 2.52646450231481981058882929207, 3.01804384251345169358926901809, 3.54450326690758281448216229050, 3.89908480426948392711504438086, 5.01640021464643343700684606261, 5.16777488930779092557932302788, 6.10107696557531263189876906822, 6.41201118200370857374720017569, 6.78682437386849717586583618209, 6.91964822574493532226298466197, 7.73015386086090233925026670194, 8.276581415230693607756604397446, 8.457880985479899014756103944006
