L(s) = 1 | + 3·2-s − 3-s + 6·4-s − 5-s − 3·6-s + 10·8-s − 3·10-s − 11-s − 6·12-s + 15-s + 15·16-s − 17-s − 6·20-s − 3·22-s − 23-s − 10·24-s − 29-s + 3·30-s + 21·32-s + 33-s − 3·34-s − 37-s − 10·40-s − 43-s − 6·44-s − 3·46-s − 15·48-s + ⋯ |
L(s) = 1 | + 3·2-s − 3-s + 6·4-s − 5-s − 3·6-s + 10·8-s − 3·10-s − 11-s − 6·12-s + 15-s + 15·16-s − 17-s − 6·20-s − 3·22-s − 23-s − 10·24-s − 29-s + 3·30-s + 21·32-s + 33-s − 3·34-s − 37-s − 10·40-s − 43-s − 6·44-s − 3·46-s − 15·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 149^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 149^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.717566146\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.717566146\) |
\(L(1)\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 149 | $C_1$ | \( ( 1 - T )^{3} \) |
good | 3 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 5 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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_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 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 73 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 79 | $C_1$ | \( ( 1 - T )^{6} \) |
| 83 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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
−10.23519609818793842012176404331, −9.501424400605764964089622521057, −9.250710631401467976121059598412, −8.767118039160490798148904336070, −8.135827248178107369002482843541, −8.038636972060988918738943736182, −7.70470240063828502591835159521, −7.54809136072691429558417224836, −7.01650201432097226932299763561, −6.93428759958540294842988685146, −6.51175436038185842829922232502, −6.09024939660040209490814059774, −5.99931011134809662402960112111, −5.57244091159838035437603336380, −5.36206990189740613182160144318, −4.98864267022803666627479342176, −4.71202605665370548558540322745, −4.39039819139761358672467783552, −3.97980067918229553706481218130, −3.68372837414797092259550574237, −3.42114643743721253675311719195, −2.92665907072410479575838754014, −2.31818143150346579226245168622, −2.17710529515360003247064022929, −1.49205100798263179875774308984,
1.49205100798263179875774308984, 2.17710529515360003247064022929, 2.31818143150346579226245168622, 2.92665907072410479575838754014, 3.42114643743721253675311719195, 3.68372837414797092259550574237, 3.97980067918229553706481218130, 4.39039819139761358672467783552, 4.71202605665370548558540322745, 4.98864267022803666627479342176, 5.36206990189740613182160144318, 5.57244091159838035437603336380, 5.99931011134809662402960112111, 6.09024939660040209490814059774, 6.51175436038185842829922232502, 6.93428759958540294842988685146, 7.01650201432097226932299763561, 7.54809136072691429558417224836, 7.70470240063828502591835159521, 8.038636972060988918738943736182, 8.135827248178107369002482843541, 8.767118039160490798148904336070, 9.250710631401467976121059598412, 9.501424400605764964089622521057, 10.23519609818793842012176404331