| L(s) = 1 | − 2·2-s + 4-s + 2·8-s − 2·11-s − 4·13-s − 4·16-s + 4·22-s − 2·25-s + 8·26-s + 2·32-s − 2·44-s − 2·47-s + 49-s + 4·50-s − 4·52-s + 2·59-s − 2·61-s + 3·64-s − 4·71-s + 8·83-s − 4·88-s + 4·94-s − 2·98-s − 2·100-s − 8·104-s − 4·118-s + 3·121-s + ⋯ |
| L(s) = 1 | − 2·2-s + 4-s + 2·8-s − 2·11-s − 4·13-s − 4·16-s + 4·22-s − 2·25-s + 8·26-s + 2·32-s − 2·44-s − 2·47-s + 49-s + 4·50-s − 4·52-s + 2·59-s − 2·61-s + 3·64-s − 4·71-s + 8·83-s − 4·88-s + 4·94-s − 2·98-s − 2·100-s − 8·104-s − 4·118-s + 3·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1146859812\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1146859812\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$ | \( ( 1 - T )^{8} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.57464503643451506350158049354, −6.04757729883037197632298842768, −5.75249424727858857979715373454, −5.73246559929876121598362187633, −5.62707758165988694428048042972, −5.02552014589658849385257045979, −5.01707368579663065211162380782, −4.91794785765943762765491910636, −4.89976226452983566752527016789, −4.52060139676886798444498877032, −4.47978808689521952107383542376, −4.14394346993804071122728535733, −3.79309597750209771877145022790, −3.54118849983135223969336000700, −3.42643589125464309742913367000, −2.93187041382565972191823300973, −2.70921419243504948675166885938, −2.59535936472220846965741026032, −2.17913657261610824920769881037, −2.02761266057789950772448838537, −2.02411153060843349419624386589, −1.64885849300425404908807287389, −1.15677034462573123909217393740, −0.43216482128570272292205214159, −0.42110517541478887869394374544,
0.42110517541478887869394374544, 0.43216482128570272292205214159, 1.15677034462573123909217393740, 1.64885849300425404908807287389, 2.02411153060843349419624386589, 2.02761266057789950772448838537, 2.17913657261610824920769881037, 2.59535936472220846965741026032, 2.70921419243504948675166885938, 2.93187041382565972191823300973, 3.42643589125464309742913367000, 3.54118849983135223969336000700, 3.79309597750209771877145022790, 4.14394346993804071122728535733, 4.47978808689521952107383542376, 4.52060139676886798444498877032, 4.89976226452983566752527016789, 4.91794785765943762765491910636, 5.01707368579663065211162380782, 5.02552014589658849385257045979, 5.62707758165988694428048042972, 5.73246559929876121598362187633, 5.75249424727858857979715373454, 6.04757729883037197632298842768, 6.57464503643451506350158049354
