| L(s) = 1 | + 2-s − 3·7-s + 13-s − 3·14-s − 17-s + 19-s + 23-s + 3·25-s + 26-s − 34-s − 37-s + 38-s + 3·41-s − 43-s + 46-s − 47-s + 6·49-s + 3·50-s − 74-s + 3·82-s − 86-s − 89-s − 3·91-s − 94-s + 97-s + 6·98-s − 101-s + ⋯ |
| L(s) = 1 | + 2-s − 3·7-s + 13-s − 3·14-s − 17-s + 19-s + 23-s + 3·25-s + 26-s − 34-s − 37-s + 38-s + 3·41-s − 43-s + 46-s − 47-s + 6·49-s + 3·50-s − 74-s + 3·82-s − 86-s − 89-s − 3·91-s − 94-s + 97-s + 6·98-s − 101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 7^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.673569517\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.673569517\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 13 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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_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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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
−8.194639787662539846992120698394, −7.49200955345238158844718442597, −7.45460073311576972681044200955, −7.13032876357758150503997391068, −7.04800402396822059485308327285, −6.61505146573026512372286022650, −6.48224018561616553590788887848, −6.31067638949904687736874133284, −5.94455202079573899479751621260, −5.90812992215451184766254597856, −5.17627977169377149822466210185, −5.14917741630748511329784524852, −5.10988168880710335591015813630, −4.38778498977442919908278049523, −4.26717849378088347253225224836, −4.14128133527663133054093708643, −3.55129377263065664685285013559, −3.43394535923676888123932731309, −3.08476604336815575014386300850, −2.88269371753958251897181709838, −2.76182950527945432747839531269, −2.23479743207033367216378145713, −1.64430122150954536375915455795, −0.940245613800488204021144562671, −0.74077099824192099508331289199,
0.74077099824192099508331289199, 0.940245613800488204021144562671, 1.64430122150954536375915455795, 2.23479743207033367216378145713, 2.76182950527945432747839531269, 2.88269371753958251897181709838, 3.08476604336815575014386300850, 3.43394535923676888123932731309, 3.55129377263065664685285013559, 4.14128133527663133054093708643, 4.26717849378088347253225224836, 4.38778498977442919908278049523, 5.10988168880710335591015813630, 5.14917741630748511329784524852, 5.17627977169377149822466210185, 5.90812992215451184766254597856, 5.94455202079573899479751621260, 6.31067638949904687736874133284, 6.48224018561616553590788887848, 6.61505146573026512372286022650, 7.04800402396822059485308327285, 7.13032876357758150503997391068, 7.45460073311576972681044200955, 7.49200955345238158844718442597, 8.194639787662539846992120698394
