| L(s) = 1 | − 4·3-s − 2·4-s + 10·9-s + 2·11-s + 8·12-s + 3·16-s − 2·17-s + 4·19-s − 25-s − 20·27-s − 8·33-s − 20·36-s + 2·41-s − 4·44-s − 12·48-s − 49-s + 8·51-s − 16·57-s − 2·59-s − 4·64-s + 8·67-s + 4·68-s − 2·73-s + 4·75-s − 8·76-s + 35·81-s + 2·83-s + ⋯ |
| L(s) = 1 | − 4·3-s − 2·4-s + 10·9-s + 2·11-s + 8·12-s + 3·16-s − 2·17-s + 4·19-s − 25-s − 20·27-s − 8·33-s − 20·36-s + 2·41-s − 4·44-s − 12·48-s − 49-s + 8·51-s − 16·57-s − 2·59-s − 4·64-s + 8·67-s + 4·68-s − 2·73-s + 4·75-s − 8·76-s + 35·81-s + 2·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2063659436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2063659436\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_1$ | \( ( 1 - T )^{8} \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−7.00975701338042060212418530383, −6.88338426545065939874180397311, −6.51697632667045725581370962330, −6.16295610399732562447420121596, −6.14046629083461239473195480549, −6.00881636040310916099743565699, −5.68533425837039125906128231297, −5.43551125392601401274057227010, −5.20759666615194740125820140503, −5.12940009761133881595293212754, −5.08414426631715580775427205765, −4.66874248408021146650973203897, −4.37402831368063362271674168739, −4.30356943788939694742619366100, −4.15116533850862566982755605435, −3.70736182725343264273357302482, −3.65820971692902584889112492813, −3.50908298697910065660798941257, −2.99455399907437500758744137668, −2.28048000810519457532227374482, −1.93328793093652259146125002360, −1.41040769341772955298021833895, −1.25703114398186467022443669450, −0.891104969671182459887282015517, −0.64883884252614936288081504103,
0.64883884252614936288081504103, 0.891104969671182459887282015517, 1.25703114398186467022443669450, 1.41040769341772955298021833895, 1.93328793093652259146125002360, 2.28048000810519457532227374482, 2.99455399907437500758744137668, 3.50908298697910065660798941257, 3.65820971692902584889112492813, 3.70736182725343264273357302482, 4.15116533850862566982755605435, 4.30356943788939694742619366100, 4.37402831368063362271674168739, 4.66874248408021146650973203897, 5.08414426631715580775427205765, 5.12940009761133881595293212754, 5.20759666615194740125820140503, 5.43551125392601401274057227010, 5.68533425837039125906128231297, 6.00881636040310916099743565699, 6.14046629083461239473195480549, 6.16295610399732562447420121596, 6.51697632667045725581370962330, 6.88338426545065939874180397311, 7.00975701338042060212418530383
