| L(s) = 1 | + 4-s − 4·5-s − 6·9-s + 16-s − 4·20-s + 16·23-s + 2·25-s − 2·31-s − 6·36-s + 20·37-s + 24·45-s − 16·47-s − 14·49-s − 12·53-s − 24·59-s + 64-s − 24·67-s + 16·71-s − 4·80-s + 27·81-s − 12·89-s + 16·92-s + 4·97-s + 2·100-s + 16·103-s + 4·113-s − 64·115-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.78·5-s − 2·9-s + 1/4·16-s − 0.894·20-s + 3.33·23-s + 2/5·25-s − 0.359·31-s − 36-s + 3.28·37-s + 3.57·45-s − 2.33·47-s − 2·49-s − 1.64·53-s − 3.12·59-s + 1/8·64-s − 2.93·67-s + 1.89·71-s − 0.447·80-s + 3·81-s − 1.27·89-s + 1.66·92-s + 0.406·97-s + 1/5·100-s + 1.57·103-s + 0.376·113-s − 5.96·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 465124 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465124 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7485291848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7485291848\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 31 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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
−8.366950675915466890526830696713, −7.972117516729530841460864476054, −7.76857292337562945056886541715, −7.44813954497716788319038433151, −6.62362054253814874759895013405, −6.32858579420662158347624019611, −5.90000984791265077671128060122, −5.12409663272380286045429532906, −4.74203912193695697715499891246, −4.30914563361016729079940273832, −3.22916369280236661872194294152, −3.16188554299090604324575557253, −2.86951068240353434231414371750, −1.61564515331644559705016592088, −0.45818161262278296890065564652,
0.45818161262278296890065564652, 1.61564515331644559705016592088, 2.86951068240353434231414371750, 3.16188554299090604324575557253, 3.22916369280236661872194294152, 4.30914563361016729079940273832, 4.74203912193695697715499891246, 5.12409663272380286045429532906, 5.90000984791265077671128060122, 6.32858579420662158347624019611, 6.62362054253814874759895013405, 7.44813954497716788319038433151, 7.76857292337562945056886541715, 7.972117516729530841460864476054, 8.366950675915466890526830696713
