| L(s) = 1 | + 2-s + 4-s − 4·5-s + 7-s − 9-s − 4·10-s + 14-s − 18-s + 19-s − 4·20-s + 6·25-s + 28-s − 32-s − 4·35-s − 36-s + 38-s + 41-s + 4·45-s − 2·47-s + 49-s + 6·50-s + 59-s − 63-s − 64-s + 8·67-s − 4·70-s + 71-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 4·5-s + 7-s − 9-s − 4·10-s + 14-s − 18-s + 19-s − 4·20-s + 6·25-s + 28-s − 32-s − 4·35-s − 36-s + 38-s + 41-s + 4·45-s − 2·47-s + 49-s + 6·50-s + 59-s − 63-s − 64-s + 8·67-s − 4·70-s + 71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6389026399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6389026399\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 31 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 3 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 7 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 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$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 41 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 43 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 47 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_1$ | \( ( 1 - T )^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 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_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 89 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_4\times C_2$ | \( 1 - T + T^{3} - T^{4} + T^{5} - T^{7} + T^{8} \) |
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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
−7.39745140265450717735596783736, −7.36202380002155801657416394872, −7.03021885067196271822906544245, −6.74687668537547120398363177976, −6.58543501692979862242916410041, −6.39132316265927062708174563964, −5.90119211391729129107191851526, −5.61640950188835646776086368368, −5.58962474294699969164112923343, −5.38459402420678534271156286873, −4.92070586494791923869689276324, −4.74563035446808683017939455733, −4.70309477190647836673756099552, −4.15823935514117878179352947406, −3.99992463075727683850497885626, −3.91402161414175759583179288958, −3.41600072498515458727613160522, −3.40132014597364513401999463594, −3.38262884136449105992825694290, −3.07408450536182380060690743409, −2.22380342541582270248557549973, −2.18994136627161153429654453173, −2.12279685446713081119735335241, −0.990534801976033516665361488151, −0.67741350491287073570256142602,
0.67741350491287073570256142602, 0.990534801976033516665361488151, 2.12279685446713081119735335241, 2.18994136627161153429654453173, 2.22380342541582270248557549973, 3.07408450536182380060690743409, 3.38262884136449105992825694290, 3.40132014597364513401999463594, 3.41600072498515458727613160522, 3.91402161414175759583179288958, 3.99992463075727683850497885626, 4.15823935514117878179352947406, 4.70309477190647836673756099552, 4.74563035446808683017939455733, 4.92070586494791923869689276324, 5.38459402420678534271156286873, 5.58962474294699969164112923343, 5.61640950188835646776086368368, 5.90119211391729129107191851526, 6.39132316265927062708174563964, 6.58543501692979862242916410041, 6.74687668537547120398363177976, 7.03021885067196271822906544245, 7.36202380002155801657416394872, 7.39745140265450717735596783736
