| L(s) = 1 | + 2·3-s − 4·5-s + 4·7-s + 9-s − 6·11-s + 4·13-s − 8·15-s − 4·17-s − 3·19-s + 8·21-s + 12·23-s + 2·29-s − 2·31-s − 12·33-s − 16·35-s − 2·37-s + 8·39-s − 4·41-s − 2·43-s − 4·45-s − 6·49-s − 8·51-s + 14·53-s + 24·55-s − 6·57-s + 28·59-s − 8·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s + 1.51·7-s + 1/3·9-s − 1.80·11-s + 1.10·13-s − 2.06·15-s − 0.970·17-s − 0.688·19-s + 1.74·21-s + 2.50·23-s + 0.371·29-s − 0.359·31-s − 2.08·33-s − 2.70·35-s − 0.328·37-s + 1.28·39-s − 0.624·41-s − 0.304·43-s − 0.596·45-s − 6/7·49-s − 1.12·51-s + 1.92·53-s + 3.23·55-s − 0.794·57-s + 3.64·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.551511145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.551511145\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | | \( 1 \) | |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} - 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ac_d_ae |
| 5 | $S_4\times C_2$ | \( 1 + 4 T + 16 T^{2} + 36 T^{3} + 16 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.e_q_bk |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 22 T^{2} - 52 T^{3} + 22 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_w_aca |
| 11 | $S_4\times C_2$ | \( 1 + 6 T + 38 T^{2} + 128 T^{3} + 38 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.g_bm_ey |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 15 T^{2} - 40 T^{3} + 15 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ae_p_abo |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 40 T^{2} + 138 T^{3} + 40 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.e_bo_fi |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) | 3.23.am_en_axs |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 33 T^{2} + 60 T^{3} + 33 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ac_bh_ci |
| 31 | $S_4\times C_2$ | \( 1 + 2 T - T^{2} - 228 T^{3} - p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.c_ab_aiu |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 97 T^{2} + 116 T^{3} + 97 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.c_dt_em |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 73 T^{2} + 396 T^{3} + 73 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.e_cv_pg |
| 43 | $S_4\times C_2$ | \( 1 + 2 T + 98 T^{2} + 216 T^{3} + 98 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.c_du_ii |
| 47 | $S_4\times C_2$ | \( 1 + 50 T^{2} - 332 T^{3} + 50 p T^{4} + p^{3} T^{6} \) | 3.47.a_by_amu |
| 53 | $S_4\times C_2$ | \( 1 - 14 T + 145 T^{2} - 1116 T^{3} + 145 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ao_fp_abqy |
| 59 | $S_4\times C_2$ | \( 1 - 28 T + 421 T^{2} - 3960 T^{3} + 421 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.abc_qf_afwi |
| 61 | $S_4\times C_2$ | \( 1 + 8 T + 108 T^{2} + 380 T^{3} + 108 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.i_ee_oq |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 75 T^{2} + 1132 T^{3} + 75 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.g_cx_bro |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 201 T^{2} - 1536 T^{3} + 201 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aq_ht_achc |
| 73 | $S_4\times C_2$ | \( 1 - 24 T + 348 T^{2} - 3458 T^{3} + 348 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ay_nk_afda |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 191 T^{2} - 964 T^{3} + 191 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ag_hj_ablc |
| 83 | $S_4\times C_2$ | \( 1 + 20 T + 285 T^{2} + 3336 T^{3} + 285 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.u_kz_eyi |
| 89 | $S_4\times C_2$ | \( 1 + 209 T^{2} + 124 T^{3} + 209 p T^{4} + p^{3} T^{6} \) | 3.89.a_ib_eu |
| 97 | $S_4\times C_2$ | \( 1 + 22 T + 435 T^{2} + 4540 T^{3} + 435 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.w_qt_gsq |
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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
−7.40661108018229965151089689135, −7.15167834638286265547870917606, −7.01274061354149307983315473686, −6.70735852721148669951675757316, −6.49470290498488940317184982866, −6.11402915162912191876630620735, −5.85903817681380072410611151779, −5.23793277569753189864460420820, −5.17552950207662865842239582383, −5.13987404462388149855918697938, −4.98185226643599062992368296691, −4.44513180218592542917121981770, −4.21545955401536147414142872139, −3.91695651045439057874594863350, −3.71328788543639739071604001659, −3.67839275411268903435773525884, −3.04246651076748596152616067843, −2.86255537437610528427755800971, −2.75555570049778777513910216076, −2.21145624622053697853736077715, −1.95983727748277663696032983082, −1.79232079303619236940324989841, −1.12474751477503475778201019196, −0.76242117372370062327323116832, −0.30542797170887196444456503367,
0.30542797170887196444456503367, 0.76242117372370062327323116832, 1.12474751477503475778201019196, 1.79232079303619236940324989841, 1.95983727748277663696032983082, 2.21145624622053697853736077715, 2.75555570049778777513910216076, 2.86255537437610528427755800971, 3.04246651076748596152616067843, 3.67839275411268903435773525884, 3.71328788543639739071604001659, 3.91695651045439057874594863350, 4.21545955401536147414142872139, 4.44513180218592542917121981770, 4.98185226643599062992368296691, 5.13987404462388149855918697938, 5.17552950207662865842239582383, 5.23793277569753189864460420820, 5.85903817681380072410611151779, 6.11402915162912191876630620735, 6.49470290498488940317184982866, 6.70735852721148669951675757316, 7.01274061354149307983315473686, 7.15167834638286265547870917606, 7.40661108018229965151089689135
