L(s) = 1 | − 18·9-s + 60·17-s − 14·25-s + 36·41-s + 98·49-s + 220·73-s + 243·81-s + 156·89-s − 260·97-s − 60·113-s − 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.08e3·153-s + 157-s + 163-s + 167-s − 238·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | − 2·9-s + 3.52·17-s − 0.559·25-s + 0.878·41-s + 2·49-s + 3.01·73-s + 3·81-s + 1.75·89-s − 2.68·97-s − 0.530·113-s − 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s − 7.05·153-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.40·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16384 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.501830137\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.501830137\) |
\(L(2)\) |
|
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 | | \( 1 \) |
good | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 6 T + p^{2} T^{2} )( 1 + 6 T + p^{2} T^{2} ) \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )( 1 + 10 T + p^{2} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 70 T + p^{2} T^{2} )( 1 + 70 T + p^{2} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 18 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )( 1 + 90 T + p^{2} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )( 1 + 22 T + p^{2} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 78 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 130 T + p^{2} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.77676078865330230683853503155, −12.60696087425977997847109937278, −12.20865951635231157041891798894, −11.96485602203701156075599325867, −11.39787022735936419902063736354, −10.76492532689609751239162206837, −10.33729263689225348720288237108, −9.584591927863906772345047956544, −9.294466469909227778540559662291, −8.481061015318365249472335382929, −7.83655585834435200093201307417, −7.82339865808919886715035214529, −6.77517785117455227179730627952, −5.77243802406384235237393007373, −5.71263577535222720521205607235, −5.11864648982598716729457242297, −3.79571626667575362714368958340, −3.26350088861955763918854450182, −2.47511284011557875413663458057, −0.885053195404485168075259896355,
0.885053195404485168075259896355, 2.47511284011557875413663458057, 3.26350088861955763918854450182, 3.79571626667575362714368958340, 5.11864648982598716729457242297, 5.71263577535222720521205607235, 5.77243802406384235237393007373, 6.77517785117455227179730627952, 7.82339865808919886715035214529, 7.83655585834435200093201307417, 8.481061015318365249472335382929, 9.294466469909227778540559662291, 9.584591927863906772345047956544, 10.33729263689225348720288237108, 10.76492532689609751239162206837, 11.39787022735936419902063736354, 11.96485602203701156075599325867, 12.20865951635231157041891798894, 12.60696087425977997847109937278, 13.77676078865330230683853503155