L(s) = 1 | − 4·3-s + 4-s + 6·9-s + 6·11-s − 4·12-s + 16-s − 10·25-s + 4·27-s − 8·31-s − 24·33-s + 6·36-s − 8·37-s + 6·44-s − 4·48-s + 2·49-s − 12·53-s + 64-s + 16·67-s + 40·75-s − 37·81-s − 12·89-s + 32·93-s + 28·97-s + 36·99-s − 10·100-s − 32·103-s + 4·108-s + ⋯ |
L(s) = 1 | − 2.30·3-s + 1/2·4-s + 2·9-s + 1.80·11-s − 1.15·12-s + 1/4·16-s − 2·25-s + 0.769·27-s − 1.43·31-s − 4.17·33-s + 36-s − 1.31·37-s + 0.904·44-s − 0.577·48-s + 2/7·49-s − 1.64·53-s + 1/8·64-s + 1.95·67-s + 4.61·75-s − 4.11·81-s − 1.27·89-s + 3.31·93-s + 2.84·97-s + 3.61·99-s − 100-s − 3.15·103-s + 0.384·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139876 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139876 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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
−9.286071988264369821451329617999, −8.624363935884874176037324123450, −8.076455834435977550787516164955, −7.36497483724733271440784650514, −6.72084452362773307312329845344, −6.65330731261113455462227374582, −5.99911855171913780844071238816, −5.67320794510931157328652310045, −5.26894992853838733045784795081, −4.56197836266380508635038039003, −3.90229547122613769308947697962, −3.34611668167428925701071579672, −2.05022911292484155356003327044, −1.25284776655083263545298048613, 0,
1.25284776655083263545298048613, 2.05022911292484155356003327044, 3.34611668167428925701071579672, 3.90229547122613769308947697962, 4.56197836266380508635038039003, 5.26894992853838733045784795081, 5.67320794510931157328652310045, 5.99911855171913780844071238816, 6.65330731261113455462227374582, 6.72084452362773307312329845344, 7.36497483724733271440784650514, 8.076455834435977550787516164955, 8.624363935884874176037324123450, 9.286071988264369821451329617999