| L(s) = 1 | − 3·3-s + 2·5-s + 6·9-s + 5·11-s + 2·13-s − 6·15-s − 7·17-s + 4·19-s + 4·23-s − 9·25-s − 10·27-s − 29-s + 5·31-s − 15·33-s + 37-s − 6·39-s + 3·41-s + 23·43-s + 12·45-s + 7·47-s − 17·49-s + 21·51-s + 10·55-s − 12·57-s + 6·59-s + 33·61-s + 4·65-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.894·5-s + 2·9-s + 1.50·11-s + 0.554·13-s − 1.54·15-s − 1.69·17-s + 0.917·19-s + 0.834·23-s − 9/5·25-s − 1.92·27-s − 0.185·29-s + 0.898·31-s − 2.61·33-s + 0.164·37-s − 0.960·39-s + 0.468·41-s + 3.50·43-s + 1.78·45-s + 1.02·47-s − 2.42·49-s + 2.94·51-s + 1.34·55-s − 1.58·57-s + 0.781·59-s + 4.22·61-s + 0.496·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{3} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.786642728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.786642728\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 - 2 T + 13 T^{2} - 18 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ac_n_as |
| 7 | $S_4\times C_2$ | \( 1 + 17 T^{2} - 2 T^{3} + 17 p T^{4} + p^{3} T^{6} \) | 3.7.a_r_ac |
| 11 | $S_4\times C_2$ | \( 1 - 5 T + 28 T^{2} - 111 T^{3} + 28 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.af_bc_aeh |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 19 T^{2} - 2 T^{3} + 19 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_t_ac |
| 17 | $S_4\times C_2$ | \( 1 + 7 T + 46 T^{2} + 237 T^{3} + 46 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.h_bu_jd |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 25 T^{2} - 154 T^{3} + 25 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ae_z_afy |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 15 T^{2} + 66 T^{3} + 15 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ae_p_co |
| 29 | $S_4\times C_2$ | \( 1 + T + 4 T^{2} + 21 T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.29.b_e_v |
| 31 | $S_4\times C_2$ | \( 1 - 5 T + 64 T^{2} - 173 T^{3} + 64 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.af_cm_agr |
| 37 | $S_4\times C_2$ | \( 1 - T + 106 T^{2} - 73 T^{3} + 106 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ab_ec_acv |
| 43 | $S_4\times C_2$ | \( 1 - 23 T + 300 T^{2} - 2387 T^{3} + 300 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ax_lo_adnv |
| 47 | $S_4\times C_2$ | \( 1 - 7 T + 142 T^{2} - 615 T^{3} + 142 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ah_fm_axr |
| 53 | $S_4\times C_2$ | \( 1 + 111 T^{2} + 20 T^{3} + 111 p T^{4} + p^{3} T^{6} \) | 3.53.a_eh_u |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 173 T^{2} - 700 T^{3} + 173 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ag_gr_abay |
| 61 | $S_4\times C_2$ | \( 1 - 33 T + 530 T^{2} - 5165 T^{3} + 530 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.abh_uk_ahqr |
| 67 | $S_4\times C_2$ | \( 1 - 20 T + 281 T^{2} - 2616 T^{3} + 281 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.au_kv_adwq |
| 71 | $S_4\times C_2$ | \( 1 - 17 T + 158 T^{2} - 1141 T^{3} + 158 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ar_gc_abrx |
| 73 | $S_4\times C_2$ | \( 1 + 9 T + 230 T^{2} + 1277 T^{3} + 230 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.j_iw_bxd |
| 79 | $S_4\times C_2$ | \( 1 + 16 T + 229 T^{2} + 2320 T^{3} + 229 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.q_iv_dlg |
| 83 | $S_4\times C_2$ | \( 1 - 18 T + 219 T^{2} - 1838 T^{3} + 219 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.as_il_acss |
| 89 | $S_4\times C_2$ | \( 1 + 18 T + 311 T^{2} + 3164 T^{3} + 311 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.s_lz_ers |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 231 T^{2} + 826 T^{3} + 231 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.e_ix_bfu |
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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
−6.95245807652058336322932277791, −6.48182559760230932500806434042, −6.36962245390386122151436966967, −6.36641890072822413622213910302, −6.07836071454543927680965535872, −5.71683047501658835790738513888, −5.59420863723375700220987708446, −5.31691207531797937341780444337, −5.19991748753320263689520576841, −4.94428609878114914144667924682, −4.47252511373456539186177996778, −4.28883718379424402518265995535, −4.18625984346459358415756522994, −3.76970126215378085672564611446, −3.70737605983191301171213564072, −3.52848151954075743140850094166, −2.78196351090092798731565742977, −2.55211654464447980760476797507, −2.36742041864020547998945484362, −2.01601242718654370030843670871, −1.64660965517345277284898385923, −1.46446354646502817046496358995, −0.794541804652170427057513530178, −0.77345905755859188386010107246, −0.53370842287831372190816108656,
0.53370842287831372190816108656, 0.77345905755859188386010107246, 0.794541804652170427057513530178, 1.46446354646502817046496358995, 1.64660965517345277284898385923, 2.01601242718654370030843670871, 2.36742041864020547998945484362, 2.55211654464447980760476797507, 2.78196351090092798731565742977, 3.52848151954075743140850094166, 3.70737605983191301171213564072, 3.76970126215378085672564611446, 4.18625984346459358415756522994, 4.28883718379424402518265995535, 4.47252511373456539186177996778, 4.94428609878114914144667924682, 5.19991748753320263689520576841, 5.31691207531797937341780444337, 5.59420863723375700220987708446, 5.71683047501658835790738513888, 6.07836071454543927680965535872, 6.36641890072822413622213910302, 6.36962245390386122151436966967, 6.48182559760230932500806434042, 6.95245807652058336322932277791
