| L(s) = 1 | + 3·2-s + 6·4-s − 7-s + 10·8-s + 3·9-s − 11-s − 3·14-s + 15·16-s − 17-s + 9·18-s − 19-s − 3·22-s − 23-s + 3·25-s − 6·28-s − 29-s − 31-s + 21·32-s − 3·34-s + 18·36-s − 37-s − 3·38-s − 41-s − 43-s − 6·44-s − 3·46-s + 9·50-s + ⋯ |
| L(s) = 1 | + 3·2-s + 6·4-s − 7-s + 10·8-s + 3·9-s − 11-s − 3·14-s + 15·16-s − 17-s + 9·18-s − 19-s − 3·22-s − 23-s + 3·25-s − 6·28-s − 29-s − 31-s + 21·32-s − 3·34-s + 18·36-s − 37-s − 3·38-s − 41-s − 43-s − 6·44-s − 3·46-s + 9·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 251^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 251^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(11.31177397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.31177397\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 251 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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_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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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_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
−8.072614514188752656720750219597, −7.82135070647735025724817902482, −7.57219883163814118285651831258, −7.18967309368170271650745755970, −6.96286850243061467356775829999, −6.85796850160243507272025522402, −6.85446611413567599809651854315, −6.35289723487539999629854009989, −6.10152828771663925452510106786, −6.09730481407786616323716315548, −5.31893453327012316495797516672, −5.13453697736513187885014052533, −5.13081514742221194470586310891, −4.68071065382005982257663483019, −4.40157628475435466194178330581, −4.26276995996199409891652902832, −3.93671003577164193694341772594, −3.63280360133122603551530594142, −3.38788895230886524496922558945, −2.85265783201785653389638142199, −2.83085693221179326680520788488, −2.30140953731774952545730097360, −1.79916743201281788443203136104, −1.51044338409629372437500213854, −1.46625051177396352223191823842,
1.46625051177396352223191823842, 1.51044338409629372437500213854, 1.79916743201281788443203136104, 2.30140953731774952545730097360, 2.83085693221179326680520788488, 2.85265783201785653389638142199, 3.38788895230886524496922558945, 3.63280360133122603551530594142, 3.93671003577164193694341772594, 4.26276995996199409891652902832, 4.40157628475435466194178330581, 4.68071065382005982257663483019, 5.13081514742221194470586310891, 5.13453697736513187885014052533, 5.31893453327012316495797516672, 6.09730481407786616323716315548, 6.10152828771663925452510106786, 6.35289723487539999629854009989, 6.85446611413567599809651854315, 6.85796850160243507272025522402, 6.96286850243061467356775829999, 7.18967309368170271650745755970, 7.57219883163814118285651831258, 7.82135070647735025724817902482, 8.072614514188752656720750219597
