| L(s) = 1 | − 2-s − 7-s + 3·9-s + 14-s − 17-s − 3·18-s − 19-s + 3·25-s − 29-s − 31-s + 34-s − 37-s + 38-s − 41-s − 43-s − 47-s − 3·50-s − 53-s + 58-s + 62-s − 3·63-s − 73-s + 74-s + 6·81-s + 82-s − 83-s + 86-s + ⋯ |
| L(s) = 1 | − 2-s − 7-s + 3·9-s + 14-s − 17-s − 3·18-s − 19-s + 3·25-s − 29-s − 31-s + 34-s − 37-s + 38-s − 41-s − 43-s − 47-s − 3·50-s − 53-s + 58-s + 62-s − 3·63-s − 73-s + 74-s + 6·81-s + 82-s − 83-s + 86-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11089567 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11089567 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2167588459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2167588459\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 223 | $C_1$ | \( ( 1 - T )^{3} \) |
| 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 41 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
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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
−11.18329213836965930447921717848, −10.65950802027544333166243679837, −10.55416614288987963905383744562, −10.18156337745373947149249969145, −9.722940180343311781798736195780, −9.647429129950972227387983364873, −9.429259243939733247704475464569, −8.750336106556182382374099230039, −8.689224023698669947990192756682, −8.590387440487660438654227127077, −7.77641995845174558804019994534, −7.46237075046893909605289006309, −7.08254042924647417596349105586, −6.78890408291012346128980758825, −6.53785526790508715358636531261, −6.47019847019950335707561114342, −5.55864927403107778436837179834, −4.99908454683128382167313154003, −4.75983059683703751674071207728, −4.30576800813932447267802524804, −3.89621014452100464226196182117, −3.37943994582094599453076502764, −2.86145620745815699929172168862, −1.82609020027369136479072618297, −1.53931753575375780239609758915,
1.53931753575375780239609758915, 1.82609020027369136479072618297, 2.86145620745815699929172168862, 3.37943994582094599453076502764, 3.89621014452100464226196182117, 4.30576800813932447267802524804, 4.75983059683703751674071207728, 4.99908454683128382167313154003, 5.55864927403107778436837179834, 6.47019847019950335707561114342, 6.53785526790508715358636531261, 6.78890408291012346128980758825, 7.08254042924647417596349105586, 7.46237075046893909605289006309, 7.77641995845174558804019994534, 8.590387440487660438654227127077, 8.689224023698669947990192756682, 8.750336106556182382374099230039, 9.429259243939733247704475464569, 9.647429129950972227387983364873, 9.722940180343311781798736195780, 10.18156337745373947149249969145, 10.55416614288987963905383744562, 10.65950802027544333166243679837, 11.18329213836965930447921717848
