| L(s) = 1 | − 8·4-s + 2·5-s − 2·7-s + 40·16-s − 2·17-s + 4·19-s − 16·20-s − 4·23-s − 10·25-s + 16·28-s − 4·29-s + 2·31-s − 4·35-s + 4·37-s − 6·41-s − 2·47-s − 10·49-s + 8·53-s + 22·59-s − 16·61-s − 160·64-s + 4·67-s + 16·68-s + 22·71-s − 22·73-s − 32·76-s − 20·79-s + ⋯ |
| L(s) = 1 | − 4·4-s + 0.894·5-s − 0.755·7-s + 10·16-s − 0.485·17-s + 0.917·19-s − 3.57·20-s − 0.834·23-s − 2·25-s + 3.02·28-s − 0.742·29-s + 0.359·31-s − 0.676·35-s + 0.657·37-s − 0.937·41-s − 0.291·47-s − 1.42·49-s + 1.09·53-s + 2.86·59-s − 2.04·61-s − 20·64-s + 0.488·67-s + 1.94·68-s + 2.61·71-s − 2.57·73-s − 3.67·76-s − 2.25·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 23^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6737944121\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6737944121\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| 29 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 2 T + 14 T^{2} - 22 T^{3} + 94 T^{4} - 22 p T^{5} + 14 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 2 T + 2 p T^{2} + 26 T^{3} + 150 T^{4} + 26 p T^{5} + 2 p^{3} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 16 T^{2} + 48 T^{3} + 94 T^{4} + 48 p T^{5} + 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 14 T^{2} - 24 T^{3} + 151 T^{4} - 24 p T^{5} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 24 T^{2} + 80 T^{3} + 593 T^{4} + 80 p T^{5} + 24 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 4 T + 20 T^{2} - 112 T^{3} + 873 T^{4} - 112 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 2 T + 86 T^{2} - 234 T^{3} + 3414 T^{4} - 234 p T^{5} + 86 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 4 T + 92 T^{2} - 336 T^{3} + 4905 T^{4} - 336 p T^{5} + 92 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 6 T + 118 T^{2} + 450 T^{3} + 5950 T^{4} + 450 p T^{5} + 118 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 80 T^{2} - 216 T^{3} + 3109 T^{4} - 216 p T^{5} + 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 2 T + 182 T^{2} + 274 T^{3} + 12694 T^{4} + 274 p T^{5} + 182 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 8 T + 156 T^{2} - 856 T^{3} + 10614 T^{4} - 856 p T^{5} + 156 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 22 T + 366 T^{2} - 4156 T^{3} + 36359 T^{4} - 4156 p T^{5} + 366 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 16 T + 252 T^{2} + 2192 T^{3} + 21062 T^{4} + 2192 p T^{5} + 252 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 4 T + 212 T^{2} - 676 T^{3} + 20262 T^{4} - 676 p T^{5} + 212 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 22 T + 414 T^{2} - 4964 T^{3} + 48903 T^{4} - 4964 p T^{5} + 414 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 22 T + 458 T^{2} + 5314 T^{3} + 56718 T^{4} + 5314 p T^{5} + 458 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 20 T + 4 p T^{2} + 3064 T^{3} + 30313 T^{4} + 3064 p T^{5} + 4 p^{3} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 10 T + 210 T^{2} - 2706 T^{3} + 21014 T^{4} - 2706 p T^{5} + 210 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 14 T + 164 T^{2} - 1808 T^{3} + 23557 T^{4} - 1808 p T^{5} + 164 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 8 T + 364 T^{2} + 2296 T^{3} + 51862 T^{4} + 2296 p T^{5} + 364 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.60036236816024595611344639706, −5.33267191890637466232042535763, −5.18005875401729618281782790652, −5.15078009086232559601775520485, −5.14444525214121841935959792255, −4.67573538329511936932759906448, −4.57752305382724680802023816326, −4.24726363542682764042234702483, −4.11866429266404922845351979212, −4.04282518831391167570972387829, −3.71951058139318957888138590555, −3.68161850689012867057199391599, −3.64343657630046690546442878745, −3.20083858574626655523413402693, −2.87277248826514824722739903050, −2.84153815633090194194589725730, −2.67007398277643549023008782042, −2.09100152541022055736364075259, −1.75699747940010784796532295062, −1.61144175001469178626127728491, −1.57099340606459063245022611898, −1.00612361074723869498029590764, −0.62463720557110214076460383818, −0.49780752502319316277501746383, −0.22920531326414077689182805904,
0.22920531326414077689182805904, 0.49780752502319316277501746383, 0.62463720557110214076460383818, 1.00612361074723869498029590764, 1.57099340606459063245022611898, 1.61144175001469178626127728491, 1.75699747940010784796532295062, 2.09100152541022055736364075259, 2.67007398277643549023008782042, 2.84153815633090194194589725730, 2.87277248826514824722739903050, 3.20083858574626655523413402693, 3.64343657630046690546442878745, 3.68161850689012867057199391599, 3.71951058139318957888138590555, 4.04282518831391167570972387829, 4.11866429266404922845351979212, 4.24726363542682764042234702483, 4.57752305382724680802023816326, 4.67573538329511936932759906448, 5.14444525214121841935959792255, 5.15078009086232559601775520485, 5.18005875401729618281782790652, 5.33267191890637466232042535763, 5.60036236816024595611344639706
