L(s) = 1 | − 2-s + 3·9-s − 17-s − 3·18-s + 3·25-s − 29-s − 31-s + 34-s − 43-s − 47-s + 3·49-s − 3·50-s + 58-s − 59-s − 61-s + 62-s − 67-s − 73-s − 79-s + 6·81-s + 86-s − 89-s + 94-s − 97-s − 3·98-s − 103-s − 109-s + ⋯ |
L(s) = 1 | − 2-s + 3·9-s − 17-s − 3·18-s + 3·25-s − 29-s − 31-s + 34-s − 43-s − 47-s + 3·49-s − 3·50-s + 58-s − 59-s − 61-s + 62-s − 67-s − 73-s − 79-s + 6·81-s + 86-s − 89-s + 94-s − 97-s − 3·98-s − 103-s − 109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99252847 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99252847 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4005516616\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4005516616\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 463 | $C_1$ | \( ( 1 - T )^{3} \) |
good | 2 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 31 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 47 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 61 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 67 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 79 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 97 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.02282119472721562010164826161, −9.690772800562192495320283739059, −9.420279414318351576008754865887, −9.309537406313710211994457605544, −8.802176876257533233106540220109, −8.683297336548361002958971666410, −8.564780281378534026423004403207, −7.77187975923572939972939538572, −7.67895468976345246391935296266, −7.19793993977303823030179476454, −7.07615270596553029970784112672, −6.85728469250161661112008094528, −6.54232569365252198298968946140, −6.12110202527078835275638678245, −5.45355563587983596896455434119, −5.34672852909576887496291364150, −4.52639380204327745514026020885, −4.47582534373088163512464677170, −4.45746725476707817177630453553, −3.64946712777233160145158521325, −3.42393146583910492731892658875, −2.70997577242153672759998014402, −2.19583316526044760104909394809, −1.41764096255587410441487095632, −1.32047019466465502659742302286,
1.32047019466465502659742302286, 1.41764096255587410441487095632, 2.19583316526044760104909394809, 2.70997577242153672759998014402, 3.42393146583910492731892658875, 3.64946712777233160145158521325, 4.45746725476707817177630453553, 4.47582534373088163512464677170, 4.52639380204327745514026020885, 5.34672852909576887496291364150, 5.45355563587983596896455434119, 6.12110202527078835275638678245, 6.54232569365252198298968946140, 6.85728469250161661112008094528, 7.07615270596553029970784112672, 7.19793993977303823030179476454, 7.67895468976345246391935296266, 7.77187975923572939972939538572, 8.564780281378534026423004403207, 8.683297336548361002958971666410, 8.802176876257533233106540220109, 9.309537406313710211994457605544, 9.420279414318351576008754865887, 9.690772800562192495320283739059, 10.02282119472721562010164826161