| L(s) = 1 | − 2·3-s + 3·4-s + 4·7-s + 2·9-s − 2·11-s − 6·12-s + 4·13-s + 5·16-s + 2·17-s + 10·19-s − 8·21-s + 6·23-s − 6·27-s + 12·28-s + 10·31-s + 4·33-s + 6·36-s − 8·39-s − 14·41-s − 2·43-s − 6·44-s − 12·47-s − 10·48-s − 2·49-s − 4·51-s + 12·52-s − 10·53-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 3/2·4-s + 1.51·7-s + 2/3·9-s − 0.603·11-s − 1.73·12-s + 1.10·13-s + 5/4·16-s + 0.485·17-s + 2.29·19-s − 1.74·21-s + 1.25·23-s − 1.15·27-s + 2.26·28-s + 1.79·31-s + 0.696·33-s + 36-s − 1.28·39-s − 2.18·41-s − 0.304·43-s − 0.904·44-s − 1.75·47-s − 1.44·48-s − 2/7·49-s − 0.560·51-s + 1.66·52-s − 1.37·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.989664123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.989664123\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
| 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
−11.59263846417408964018128818610, −11.49877995026866319420390841576, −11.01293290014164402462558776367, −10.82669726838418579362701549665, −10.05127028613271994248561309330, −9.873443212325966159410298061836, −9.050008011406213315356624561879, −8.253305675688649284095761861164, −7.87155574364968762775767305140, −7.67109668290368276829886412309, −6.75376157360806958794585756715, −6.68892470576552616216444542324, −5.92458878441165171090142572011, −5.35802131020787838166673963893, −5.10467461181155124744900307917, −4.50679979538928923128421638146, −3.20387542892187851939078231485, −3.05605240781393840939190521912, −1.56913327172388475004175740625, −1.38698629632226783143308340965,
1.38698629632226783143308340965, 1.56913327172388475004175740625, 3.05605240781393840939190521912, 3.20387542892187851939078231485, 4.50679979538928923128421638146, 5.10467461181155124744900307917, 5.35802131020787838166673963893, 5.92458878441165171090142572011, 6.68892470576552616216444542324, 6.75376157360806958794585756715, 7.67109668290368276829886412309, 7.87155574364968762775767305140, 8.253305675688649284095761861164, 9.050008011406213315356624561879, 9.873443212325966159410298061836, 10.05127028613271994248561309330, 10.82669726838418579362701549665, 11.01293290014164402462558776367, 11.49877995026866319420390841576, 11.59263846417408964018128818610
