| L(s) = 1 | + 2·2-s + 3-s − 4-s + 2·6-s − 8·8-s + 9-s − 4·11-s − 12-s − 7·16-s + 12·17-s + 2·18-s − 8·22-s − 8·24-s − 6·25-s + 27-s + 4·29-s + 14·32-s − 4·33-s + 24·34-s − 36-s + 12·37-s − 4·41-s + 4·44-s − 7·48-s + 49-s − 12·50-s + 12·51-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s − 1/2·4-s + 0.816·6-s − 2.82·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 7/4·16-s + 2.91·17-s + 0.471·18-s − 1.70·22-s − 1.63·24-s − 6/5·25-s + 0.192·27-s + 0.742·29-s + 2.47·32-s − 0.696·33-s + 4.11·34-s − 1/6·36-s + 1.97·37-s − 0.624·41-s + 0.603·44-s − 1.01·48-s + 1/7·49-s − 1.69·50-s + 1.68·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160083 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160083 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.542844193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.542844193\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 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
−9.457705101580370517739336150901, −8.699165629902587420835899742101, −8.166185648107227108757068218850, −7.81295430367747747098376616892, −7.59860530511322577039011534740, −6.50104795223760620212837147128, −5.94607665445287604157300008429, −5.62767376699147437191457045340, −5.02709416296405066539370067618, −4.79135217072233780435269822604, −3.91805277407899456693617223448, −3.56599684396412385637222444746, −3.05422074105458389777226041971, −2.43470322830479280229086031798, −0.860569456173605054070248949719,
0.860569456173605054070248949719, 2.43470322830479280229086031798, 3.05422074105458389777226041971, 3.56599684396412385637222444746, 3.91805277407899456693617223448, 4.79135217072233780435269822604, 5.02709416296405066539370067618, 5.62767376699147437191457045340, 5.94607665445287604157300008429, 6.50104795223760620212837147128, 7.59860530511322577039011534740, 7.81295430367747747098376616892, 8.166185648107227108757068218850, 8.699165629902587420835899742101, 9.457705101580370517739336150901
