| L(s) = 1 | + 2·3-s + 4-s + 9-s + 2·12-s + 4·13-s − 2·25-s − 2·27-s + 36-s + 6·37-s + 8·39-s − 2·43-s + 2·49-s + 4·52-s − 64-s − 4·75-s − 4·79-s − 4·81-s − 2·100-s − 2·108-s + 12·111-s + 4·117-s − 121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2·3-s + 4-s + 9-s + 2·12-s + 4·13-s − 2·25-s − 2·27-s + 36-s + 6·37-s + 8·39-s − 2·43-s + 2·49-s + 4·52-s − 64-s − 4·75-s − 4·79-s − 4·81-s − 2·100-s − 2·108-s + 12·111-s + 4·117-s − 121-s + 127-s − 4·129-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.737734669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.737734669\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−7.09571907216870890942660305614, −6.67085040090931358860414144844, −6.30630499702309665938167186923, −6.30147937369691704182802415681, −6.06959555173148651715270340916, −5.96288141344355515965707928039, −5.80709278743815082680386530166, −5.57842945439289366493766661320, −5.49130417752778000527620131425, −4.70240415410724919191751222982, −4.65560123037762148718833491252, −4.40163434110818350839373503107, −3.96570368644413284660659492057, −3.95628963074027665640315371396, −3.72485575481078228146443938005, −3.53948554509852849544682552848, −3.33423933166108371963427203566, −2.87438392119721906576580221641, −2.72075417532501447626349524381, −2.53243022209966109439006292743, −2.27762895805750125527464955583, −2.06690223925445867820585087038, −1.36611514249903159304354701171, −1.36274078236633075997521077275, −1.14443759536289464974120539366,
1.14443759536289464974120539366, 1.36274078236633075997521077275, 1.36611514249903159304354701171, 2.06690223925445867820585087038, 2.27762895805750125527464955583, 2.53243022209966109439006292743, 2.72075417532501447626349524381, 2.87438392119721906576580221641, 3.33423933166108371963427203566, 3.53948554509852849544682552848, 3.72485575481078228146443938005, 3.95628963074027665640315371396, 3.96570368644413284660659492057, 4.40163434110818350839373503107, 4.65560123037762148718833491252, 4.70240415410724919191751222982, 5.49130417752778000527620131425, 5.57842945439289366493766661320, 5.80709278743815082680386530166, 5.96288141344355515965707928039, 6.06959555173148651715270340916, 6.30147937369691704182802415681, 6.30630499702309665938167186923, 6.67085040090931358860414144844, 7.09571907216870890942660305614
