| 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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.684680390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684680390\) |
| \(L(2)\) |
|
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
−11.38341182990168128216609776822, −10.91502636543359360975893234589, −10.57676282819587240169114653849, −9.995807900446350032472476256431, −9.854217296909285194459277376297, −9.271686117580921966436130774485, −9.007751727035655767094991427293, −8.491423163455395543887596065900, −7.77050590704299318437566350568, −7.60274418522020420947650050760, −6.84660815991702511611277764692, −6.19978390460462349539228098792, −5.82987594022744100465275374225, −5.14298408026605656423172404619, −4.90727822443623711545050827219, −3.60046653250209872314013993422, −3.41883690034154622110848871809, −1.98516170305361835744857367911, −1.74397715522564305210952759904, −0.76484877705502247109661328735,
0.76484877705502247109661328735, 1.74397715522564305210952759904, 1.98516170305361835744857367911, 3.41883690034154622110848871809, 3.60046653250209872314013993422, 4.90727822443623711545050827219, 5.14298408026605656423172404619, 5.82987594022744100465275374225, 6.19978390460462349539228098792, 6.84660815991702511611277764692, 7.60274418522020420947650050760, 7.77050590704299318437566350568, 8.491423163455395543887596065900, 9.007751727035655767094991427293, 9.271686117580921966436130774485, 9.854217296909285194459277376297, 9.995807900446350032472476256431, 10.57676282819587240169114653849, 10.91502636543359360975893234589, 11.38341182990168128216609776822
