| L(s) = 1 | − 3·3-s − 2·5-s + 6·9-s − 4·11-s − 3·13-s + 6·15-s + 6·17-s + 25-s − 10·27-s − 6·29-s − 8·31-s + 12·33-s − 2·37-s + 9·39-s + 10·41-s − 4·43-s − 12·45-s − 8·47-s − 5·49-s − 18·51-s − 14·53-s + 8·55-s − 20·59-s − 2·61-s + 6·65-s + 16·67-s − 8·71-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.894·5-s + 2·9-s − 1.20·11-s − 0.832·13-s + 1.54·15-s + 1.45·17-s + 1/5·25-s − 1.92·27-s − 1.11·29-s − 1.43·31-s + 2.08·33-s − 0.328·37-s + 1.44·39-s + 1.56·41-s − 0.609·43-s − 1.78·45-s − 1.16·47-s − 5/7·49-s − 2.52·51-s − 1.92·53-s + 1.07·55-s − 2.60·59-s − 0.256·61-s + 0.744·65-s + 1.95·67-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 5 | $D_{6}$ | \( 1 + 2 T + 3 T^{2} + 12 T^{3} + 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.c_d_m |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.7.a_f_aq |
| 11 | $S_4\times C_2$ | \( 1 + 4 T + 17 T^{2} + 56 T^{3} + 17 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.e_r_ce |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.17.ag_cl_aie |
| 19 | $S_4\times C_2$ | \( 1 + 41 T^{2} - 16 T^{3} + 41 p T^{4} + p^{3} T^{6} \) | 3.19.a_bp_aq |
| 23 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 128 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.23.a_f_ey |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.29.g_dv_ns |
| 31 | $S_4\times C_2$ | \( 1 + 8 T + 61 T^{2} + 224 T^{3} + 61 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.i_cj_iq |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 59 T^{2} + 108 T^{3} + 59 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.c_ch_ee |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 143 T^{2} - 812 T^{3} + 143 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ak_fn_abfg |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 280 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.e_bx_ku |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 720 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.i_fl_bbs |
| 53 | $S_4\times C_2$ | \( 1 + 14 T + 171 T^{2} + 1332 T^{3} + 171 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.o_gp_bzg |
| 59 | $S_4\times C_2$ | \( 1 + 20 T + 273 T^{2} + 2328 T^{3} + 273 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.u_kn_dlo |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 35 T^{2} + 780 T^{3} + 35 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.c_bj_bea |
| 67 | $S_4\times C_2$ | \( 1 - 16 T + 137 T^{2} - 1104 T^{3} + 137 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.aq_fh_abqm |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 197 T^{2} + 976 T^{3} + 197 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.i_hp_blo |
| 73 | $S_4\times C_2$ | \( 1 - 22 T + 327 T^{2} - 3220 T^{3} + 327 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aw_mp_aetw |
| 79 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{3} \) | 3.79.bk_zt_kzg |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 217 T^{2} + 696 T^{3} + 217 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.e_ij_bau |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 159 T^{2} + 852 T^{3} + 159 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.g_gd_bgu |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 239 T^{2} + 348 T^{3} + 239 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.c_jf_nk |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.208138680530873189001335521988, −7.87440418335442811317284531771, −7.65568765266482194228999677883, −7.56182414531034302117295597647, −7.46811940351478366761976041663, −6.81127079928466563255113722937, −6.80075455446340627457320481624, −6.58740827575910643290920614795, −6.16205400882565496900175144988, −5.82100284392745922898340804404, −5.54366524145404682730752628424, −5.46775019521849160238587717867, −5.24008184038767889350456544834, −4.82453990846517455221014468214, −4.70610189358936067928252179083, −4.42856692000008426332267828617, −3.85686034813698352606770613828, −3.77748700678409020204928232764, −3.51813069576890035620238010096, −2.98665050099290651818415851415, −2.64658063535491368094320030747, −2.48668498200180421614130613944, −1.55826475508917999923602099758, −1.55104743322943519640342904714, −1.16445630478652635255946783110, 0, 0, 0,
1.16445630478652635255946783110, 1.55104743322943519640342904714, 1.55826475508917999923602099758, 2.48668498200180421614130613944, 2.64658063535491368094320030747, 2.98665050099290651818415851415, 3.51813069576890035620238010096, 3.77748700678409020204928232764, 3.85686034813698352606770613828, 4.42856692000008426332267828617, 4.70610189358936067928252179083, 4.82453990846517455221014468214, 5.24008184038767889350456544834, 5.46775019521849160238587717867, 5.54366524145404682730752628424, 5.82100284392745922898340804404, 6.16205400882565496900175144988, 6.58740827575910643290920614795, 6.80075455446340627457320481624, 6.81127079928466563255113722937, 7.46811940351478366761976041663, 7.56182414531034302117295597647, 7.65568765266482194228999677883, 7.87440418335442811317284531771, 8.208138680530873189001335521988
