| L(s) = 1 | + 4·3-s − 8·7-s + 4·9-s + 10·16-s − 8·19-s − 32·21-s − 4·27-s − 8·31-s + 4·37-s + 40·48-s + 32·49-s − 32·57-s − 56·61-s − 32·63-s − 8·67-s + 28·73-s + 16·79-s − 10·81-s − 32·93-s − 32·97-s + 64·109-s + 16·111-s − 80·112-s + 127-s + 131-s + 64·133-s + 137-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 3.02·7-s + 4/3·9-s + 5/2·16-s − 1.83·19-s − 6.98·21-s − 0.769·27-s − 1.43·31-s + 0.657·37-s + 5.77·48-s + 32/7·49-s − 4.23·57-s − 7.17·61-s − 4.03·63-s − 0.977·67-s + 3.27·73-s + 1.80·79-s − 1.11·81-s − 3.31·93-s − 3.24·97-s + 6.13·109-s + 1.51·111-s − 7.55·112-s + 0.0887·127-s + 0.0873·131-s + 5.54·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1055936627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1055936627\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( 1 \) |
| good | 2 | \( ( 1 - 5 T^{4} + p^{4} T^{8} )^{2} \) |
| 5 | \( 1 - 22 T^{4} + 939 T^{8} - 22 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 11 | \( 1 - 100 T^{4} + 4134 T^{8} - 100 p^{4} T^{12} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 38 T^{2} + 831 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + 4 T + 8 T^{2} + 60 T^{3} + 434 T^{4} + 60 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + p T^{2} )^{8} \) |
| 29 | \( ( 1 - 34 T^{2} + 1863 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 + 4 T + 8 T^{2} + 36 T^{3} - 322 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 2 T + 2 T^{2} - p^{2} T^{4} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 + 2474 T^{4} + 2947659 T^{8} + 2474 p^{4} T^{12} + p^{8} T^{16} \) |
| 43 | \( ( 1 - 100 T^{2} + 5226 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 - 5500 T^{4} + 14557062 T^{8} - 5500 p^{4} T^{12} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 190 T^{2} + 14535 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 + 6980 T^{4} + 34417254 T^{8} + 6980 p^{4} T^{12} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 7 T + p T^{2} )^{8} \) |
| 67 | \( ( 1 + 4 T + 8 T^{2} + 60 T^{3} - 2254 T^{4} + 60 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 + 11684 T^{4} + 81904134 T^{8} + 11684 p^{4} T^{12} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 14 T + 98 T^{2} - 1176 T^{3} + 13991 T^{4} - 1176 p T^{5} + 98 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 2 T + p T^{2} )^{8} \) |
| 83 | \( 1 + 21212 T^{4} + 202731366 T^{8} + 21212 p^{4} T^{12} + p^{8} T^{16} \) |
| 89 | \( 1 - 17476 T^{4} + 200482374 T^{8} - 17476 p^{4} T^{12} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 16 T + 128 T^{2} + 1200 T^{3} + 10766 T^{4} + 1200 p T^{5} + 128 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.83577485719117528973645449524, −4.56750075553390732978128345481, −4.49409294684414172859055337584, −4.28333711099501648949590123770, −4.24048787821216242431788172293, −3.80138962565327556255201378034, −3.75180697606080574781215645418, −3.65424075026996709634827584185, −3.52456466693029530013093430167, −3.39231434571532344651874479129, −3.39080671197382563548933872231, −3.19355799776355652363837878802, −2.97219060780929974974974115437, −2.96251108447483212344760010100, −2.83687476844592739171193366044, −2.47479935761714554591934009625, −2.43264035447499391738041254986, −2.19243624944762608849094562519, −2.17094398784011755504368872773, −1.86409971757795049891123667437, −1.51412910589809620798916110067, −1.27522687636798636771967110223, −1.16449008023355979786645434135, −0.58545186054593009240629220487, −0.05337998788579036836607755027,
0.05337998788579036836607755027, 0.58545186054593009240629220487, 1.16449008023355979786645434135, 1.27522687636798636771967110223, 1.51412910589809620798916110067, 1.86409971757795049891123667437, 2.17094398784011755504368872773, 2.19243624944762608849094562519, 2.43264035447499391738041254986, 2.47479935761714554591934009625, 2.83687476844592739171193366044, 2.96251108447483212344760010100, 2.97219060780929974974974115437, 3.19355799776355652363837878802, 3.39080671197382563548933872231, 3.39231434571532344651874479129, 3.52456466693029530013093430167, 3.65424075026996709634827584185, 3.75180697606080574781215645418, 3.80138962565327556255201378034, 4.24048787821216242431788172293, 4.28333711099501648949590123770, 4.49409294684414172859055337584, 4.56750075553390732978128345481, 4.83577485719117528973645449524
Plot not available for L-functions of degree greater than 10.
