| L(s) = 1 | − 3-s − 7-s − 4·9-s − 11·11-s + 5·13-s + 12·17-s − 12·19-s + 21-s + 12·23-s − 15·25-s + 6·27-s − 2·29-s + 10·31-s + 11·33-s + 2·37-s − 5·39-s + 2·41-s − 5·43-s − 19·47-s − 12·49-s − 12·51-s + 14·53-s + 12·57-s + 5·59-s + 6·61-s + 4·63-s − 35·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 4/3·9-s − 3.31·11-s + 1.38·13-s + 2.91·17-s − 2.75·19-s + 0.218·21-s + 2.50·23-s − 3·25-s + 1.15·27-s − 0.371·29-s + 1.79·31-s + 1.91·33-s + 0.328·37-s − 0.800·39-s + 0.312·41-s − 0.762·43-s − 2.77·47-s − 1.71·49-s − 1.68·51-s + 1.92·53-s + 1.58·57-s + 0.650·59-s + 0.768·61-s + 0.503·63-s − 4.27·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 503^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 503^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.224446396\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224446396\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 503 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T + 5 T^{2} + p T^{3} + 5 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 7 | $S_4\times C_2$ | \( 1 + T + 13 T^{2} + 11 T^{3} + 13 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + p T + 69 T^{2} + 277 T^{3} + 69 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 37 T^{2} - 105 T^{3} + 37 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 12 T + 79 T^{2} - 368 T^{3} + 79 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 47 T^{2} + 188 T^{3} + 47 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 10 T + 3 p T^{2} - 548 T^{3} + 3 p^{2} T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 43 T^{2} - 204 T^{3} + 43 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 19 T^{2} - 284 T^{3} + 19 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 5 T + 81 T^{2} + 435 T^{3} + 81 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 19 T + 237 T^{2} + 1849 T^{3} + 237 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 127 T^{2} - 836 T^{3} + 127 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 5 T + 161 T^{2} - 591 T^{3} + 161 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 67 | $S_4\times C_2$ | \( 1 + 35 T + 589 T^{2} + 6009 T^{3} + 589 p T^{4} + 35 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 T + 241 T^{2} + 1712 T^{3} + 241 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 35 T + 605 T^{2} - 6403 T^{3} + 605 p T^{4} - 35 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 7 T + 85 T^{2} + 39 T^{3} + 85 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - T + 241 T^{2} - 163 T^{3} + 241 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 4 T + 175 T^{2} - 880 T^{3} + 175 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 23 T + 329 T^{2} + 3379 T^{3} + 329 p T^{4} + 23 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.76252864029468929820403850683, −6.58236743061972577659759302680, −6.41178520515374211664210821169, −6.27996132661791284930389219127, −5.73127362530942931828000425507, −5.68602666704623842089066115112, −5.66187821473602796440339282645, −5.26433961887445022132788178078, −5.25182491162867611143591245849, −4.98939746765613461882751899549, −4.48409079970118505397411819494, −4.27587437579025203946525552474, −4.24935383617152101230494054732, −3.56438597147690851387005176642, −3.32198093938506394316147443465, −3.31549217843666884344043211817, −2.89400117407419641812794833949, −2.68880451136526008451990569556, −2.68173229077074173202137476147, −2.00099459223969830669223057506, −1.77824124133720069328380313235, −1.57025990380170413415093403700, −0.865391021791085777325960859578, −0.45715724949283147624762282830, −0.34001465598641251835470733566,
0.34001465598641251835470733566, 0.45715724949283147624762282830, 0.865391021791085777325960859578, 1.57025990380170413415093403700, 1.77824124133720069328380313235, 2.00099459223969830669223057506, 2.68173229077074173202137476147, 2.68880451136526008451990569556, 2.89400117407419641812794833949, 3.31549217843666884344043211817, 3.32198093938506394316147443465, 3.56438597147690851387005176642, 4.24935383617152101230494054732, 4.27587437579025203946525552474, 4.48409079970118505397411819494, 4.98939746765613461882751899549, 5.25182491162867611143591245849, 5.26433961887445022132788178078, 5.66187821473602796440339282645, 5.68602666704623842089066115112, 5.73127362530942931828000425507, 6.27996132661791284930389219127, 6.41178520515374211664210821169, 6.58236743061972577659759302680, 6.76252864029468929820403850683
