| L(s) = 1 | + 2·2-s − 8·5-s − 8·8-s − 16·10-s + 10·13-s − 16·16-s − 32·17-s + 25·25-s + 20·26-s + 40·29-s − 64·34-s − 140·37-s + 64·40-s − 80·41-s − 49·49-s + 50·50-s + 112·53-s + 80·58-s + 22·61-s + 64·64-s − 80·65-s + 220·73-s − 280·74-s + 128·80-s − 160·82-s + 256·85-s − 320·89-s + ⋯ |
| L(s) = 1 | + 2-s − 8/5·5-s − 8-s − 8/5·10-s + 0.769·13-s − 16-s − 1.88·17-s + 25-s + 0.769·26-s + 1.37·29-s − 1.88·34-s − 3.78·37-s + 8/5·40-s − 1.95·41-s − 49-s + 50-s + 2.11·53-s + 1.37·58-s + 0.360·61-s + 64-s − 1.23·65-s + 3.01·73-s − 3.78·74-s + 8/5·80-s − 1.95·82-s + 3.01·85-s − 3.59·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9182546808\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9182546808\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T + 39 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 10 T - 69 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 40 T + 759 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 80 T + 4719 T^{2} + 80 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 22 T - 3237 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 110 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )( 1 + p T + p^{2} T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 160 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 130 T + 7491 T^{2} - 130 p^{2} T^{3} + p^{4} T^{4} \) |
| 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
−11.98636428030059413797197522445, −11.12027530804224371655854997908, −11.11829084628877236405602404271, −10.26350244793926142092367740105, −9.985312261069355399978262221877, −8.870837503566441489089058178462, −8.794368793698016836283302189183, −8.444717379838500010018585427938, −7.947767147872384359725612164932, −6.97973803770808077768758938712, −6.81358686711262502366751244350, −6.41472933347097982395915889757, −5.41921783526817406563375777299, −5.03819507206687820226401091361, −4.50801195833676593463465174585, −3.75164624348903047542013367409, −3.71672629168417051970807388874, −2.90429620806351092847395531762, −1.89116904659612649578775882992, −0.37919183467176667099697283802,
0.37919183467176667099697283802, 1.89116904659612649578775882992, 2.90429620806351092847395531762, 3.71672629168417051970807388874, 3.75164624348903047542013367409, 4.50801195833676593463465174585, 5.03819507206687820226401091361, 5.41921783526817406563375777299, 6.41472933347097982395915889757, 6.81358686711262502366751244350, 6.97973803770808077768758938712, 7.947767147872384359725612164932, 8.444717379838500010018585427938, 8.794368793698016836283302189183, 8.870837503566441489089058178462, 9.985312261069355399978262221877, 10.26350244793926142092367740105, 11.11829084628877236405602404271, 11.12027530804224371655854997908, 11.98636428030059413797197522445
