| L(s) = 1 | + 2·3-s + 4·4-s + 9-s + 8·12-s + 10·16-s − 2·25-s − 2·27-s + 4·36-s − 2·43-s + 20·48-s + 49-s − 2·61-s + 20·64-s − 4·75-s + 4·79-s − 4·81-s − 8·100-s − 2·103-s − 8·108-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 10·144-s + 2·147-s + ⋯ |
| L(s) = 1 | + 2·3-s + 4·4-s + 9-s + 8·12-s + 10·16-s − 2·25-s − 2·27-s + 4·36-s − 2·43-s + 20·48-s + 49-s − 2·61-s + 20·64-s − 4·75-s + 4·79-s − 4·81-s − 8·100-s − 2·103-s − 8·108-s − 2·121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + 10·144-s + 2·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(11.48426278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.48426278\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−6.38732404658860027379510995027, −6.03915839678772079992509607185, −5.95249458955596928243037636377, −5.90163661378742136935363834672, −5.52867993822262665495069683566, −5.35363779109168185685981709906, −5.13001649403389599936093179948, −5.07343058786051468287813564611, −4.61730381879192338418289953927, −4.10490824913838260422917459534, −4.08606935045248576255948312038, −3.72640027700614220744431703369, −3.56977397857804554955629811244, −3.45935008746000476963276038698, −3.30505441354777963907830174650, −2.95093374238459842463127869838, −2.75829198716648536403476311878, −2.63836418199614725463343411497, −2.46498373550549570187149213201, −2.14375140428819996843764377047, −1.96862948860913271434557356564, −1.83657204359243327401382376216, −1.47581233420672987487139748645, −1.44758545457071734627694642464, −0.815561257211499001148503754492,
0.815561257211499001148503754492, 1.44758545457071734627694642464, 1.47581233420672987487139748645, 1.83657204359243327401382376216, 1.96862948860913271434557356564, 2.14375140428819996843764377047, 2.46498373550549570187149213201, 2.63836418199614725463343411497, 2.75829198716648536403476311878, 2.95093374238459842463127869838, 3.30505441354777963907830174650, 3.45935008746000476963276038698, 3.56977397857804554955629811244, 3.72640027700614220744431703369, 4.08606935045248576255948312038, 4.10490824913838260422917459534, 4.61730381879192338418289953927, 5.07343058786051468287813564611, 5.13001649403389599936093179948, 5.35363779109168185685981709906, 5.52867993822262665495069683566, 5.90163661378742136935363834672, 5.95249458955596928243037636377, 6.03915839678772079992509607185, 6.38732404658860027379510995027
