| L(s) = 1 | + 4-s + 4·5-s − 11-s + 4·20-s + 10·25-s + 2·31-s − 44-s − 4·49-s − 4·55-s − 3·59-s − 5·67-s + 2·71-s − 2·89-s + 10·100-s + 5·103-s + 2·124-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·155-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 4-s + 4·5-s − 11-s + 4·20-s + 10·25-s + 2·31-s − 44-s − 4·49-s − 4·55-s − 3·59-s − 5·67-s + 2·71-s − 2·89-s + 10·100-s + 5·103-s + 2·124-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 8·155-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(4.663372372\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.663372372\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 17 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 61 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 71 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 73 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 89 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.32911793190286457739607158090, −6.27342181071403699703436872744, −6.23103507937097551479033696819, −6.07506670182930609989738495796, −5.91376023372445263886325437498, −5.44266497999937948195739653293, −5.30483549549820423378329445341, −5.15460359929798450303782351605, −5.07458094220880641379830646995, −4.74864224068381639712355233430, −4.47302120390318565085664344429, −4.44324285625666295640229763125, −4.26614063563971840183927080377, −3.41174870950071968561758765262, −3.25285464978391078147813249097, −3.13714435670354636173037871298, −2.94670272817731602093583741785, −2.75601856410794500828758906981, −2.47107315040799244684875238307, −2.13151234307518238148466551844, −2.11915385345619457948981754322, −1.61369435152524781883781148554, −1.48337243827099573047645183010, −1.46985105691505807431284062905, −0.852950772301929920977577402153,
0.852950772301929920977577402153, 1.46985105691505807431284062905, 1.48337243827099573047645183010, 1.61369435152524781883781148554, 2.11915385345619457948981754322, 2.13151234307518238148466551844, 2.47107315040799244684875238307, 2.75601856410794500828758906981, 2.94670272817731602093583741785, 3.13714435670354636173037871298, 3.25285464978391078147813249097, 3.41174870950071968561758765262, 4.26614063563971840183927080377, 4.44324285625666295640229763125, 4.47302120390318565085664344429, 4.74864224068381639712355233430, 5.07458094220880641379830646995, 5.15460359929798450303782351605, 5.30483549549820423378329445341, 5.44266497999937948195739653293, 5.91376023372445263886325437498, 6.07506670182930609989738495796, 6.23103507937097551479033696819, 6.27342181071403699703436872744, 6.32911793190286457739607158090
