| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{15} \cdot 3^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.377684889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.377684889\) |
| \(L(\frac{3}{2})\) |
|
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.ac_d_am |
| 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_q |
| 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_aey |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.29.ag_dv_ans |
| 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.ai_cj_aiq |
| 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.ac_ch_aee |
| 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.ai_fl_abbs |
| 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.ao_gp_abzg |
| 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.ac_bj_abea |
| 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.ai_hp_ablo |
| 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.abk_zt_akzg |
| 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 |
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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
−8.517800766796481598436269093152, −8.193846053361744614466619129454, −8.040938357485383457994899069343, −7.896180746913405507447611433303, −7.48387630450277437107113140678, −7.18890579725124698173710425059, −6.82937993735244516798632750862, −6.36340797076131913875515841084, −6.35504145816321506991078193235, −6.23711812070079829379982267602, −5.62421876259807927385982633257, −5.52590387319906105452560184530, −5.39155252434833116677710501604, −4.89574292149991508476300200152, −4.75870903709063069451579861391, −4.44074000228001111749730067190, −3.91778161693034905165685159112, −3.55467812403799313169669302550, −3.32251210244946787787970140738, −2.67591872543313035699312406857, −2.35333955724718377733505261174, −2.06966430075511182920492451350, −1.32014138249431687585084996141, −0.810703796064006188286456993622, −0.73211253131823077536198210954,
0.73211253131823077536198210954, 0.810703796064006188286456993622, 1.32014138249431687585084996141, 2.06966430075511182920492451350, 2.35333955724718377733505261174, 2.67591872543313035699312406857, 3.32251210244946787787970140738, 3.55467812403799313169669302550, 3.91778161693034905165685159112, 4.44074000228001111749730067190, 4.75870903709063069451579861391, 4.89574292149991508476300200152, 5.39155252434833116677710501604, 5.52590387319906105452560184530, 5.62421876259807927385982633257, 6.23711812070079829379982267602, 6.35504145816321506991078193235, 6.36340797076131913875515841084, 6.82937993735244516798632750862, 7.18890579725124698173710425059, 7.48387630450277437107113140678, 7.896180746913405507447611433303, 8.040938357485383457994899069343, 8.193846053361744614466619129454, 8.517800766796481598436269093152
