| L(s) = 1 | − 6·5-s − 2·7-s − 2·11-s + 8·17-s − 4·19-s + 8·23-s + 19·25-s − 2·29-s + 12·31-s + 12·35-s + 12·37-s + 6·41-s + 4·43-s + 2·47-s − 11·49-s − 16·53-s + 12·55-s + 36·59-s + 6·61-s + 2·67-s − 6·71-s − 6·73-s + 4·77-s + 12·79-s + 28·83-s − 48·85-s + 8·89-s + ⋯ |
| L(s) = 1 | − 2.68·5-s − 0.755·7-s − 0.603·11-s + 1.94·17-s − 0.917·19-s + 1.66·23-s + 19/5·25-s − 0.371·29-s + 2.15·31-s + 2.02·35-s + 1.97·37-s + 0.937·41-s + 0.609·43-s + 0.291·47-s − 1.57·49-s − 2.19·53-s + 1.61·55-s + 4.68·59-s + 0.768·61-s + 0.244·67-s − 0.712·71-s − 0.702·73-s + 0.455·77-s + 1.35·79-s + 3.07·83-s − 5.20·85-s + 0.847·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4072966656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4072966656\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) | |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.g_r_cc_ga |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 6 T^{2} - 24 T^{3} - 65 T^{4} - 24 p T^{5} - 6 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.c_ag_ay_acn |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ai_co_amy_cpf |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.19.e_aba_q_bqh |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 37 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.ai_dm_asm_err |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 3 T^{2} - 102 T^{3} - 908 T^{4} - 102 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.c_ad_ady_abiy |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 59 T^{2} - 276 T^{3} + 1800 T^{4} - 276 p T^{5} + 59 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.am_ch_akq_crg |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 31 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.am_du_abfk_jkt |
| 41 | $C_2^2$ | \( ( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.ag_acd_acc_hpc |
| 43 | $D_4\times C_2$ | \( 1 - 4 T - 61 T^{2} + 36 T^{3} + 3392 T^{4} + 36 p T^{5} - 61 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.ae_acj_bk_fam |
| 47 | $D_4\times C_2$ | \( 1 - 2 T - 39 T^{2} + 102 T^{3} - 548 T^{4} + 102 p T^{5} - 39 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ac_abn_dy_avc |
| 53 | $D_4\times C_2$ | \( 1 + 16 T + 99 T^{2} + 816 T^{3} + 8272 T^{4} + 816 p T^{5} + 99 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.q_dv_bfk_mge |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) | 4.59.abk_bbu_antg_evlj |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 82 T^{2} + 24 T^{3} + 8007 T^{4} + 24 p T^{5} - 82 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ag_ade_y_lvz |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 14 T^{2} + 232 T^{3} - 4433 T^{4} + 232 p T^{5} - 14 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.ac_ao_iy_agon |
| 71 | $C_2^2$ | \( ( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.g_ael_cc_wmu |
| 73 | $D_4\times C_2$ | \( 1 + 6 T - 67 T^{2} - 258 T^{3} + 2652 T^{4} - 258 p T^{5} - 67 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.g_acp_ajy_dya |
| 79 | $D_4\times C_2$ | \( 1 - 12 T - 37 T^{2} - 276 T^{3} + 15144 T^{4} - 276 p T^{5} - 37 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.am_abl_akq_wkm |
| 83 | $D_{4}$ | \( ( 1 - 14 T + 202 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.abc_xc_aluy_eyws |
| 89 | $D_{4}$ | \( ( 1 - 4 T + 169 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ai_nq_adbk_crxj |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 29 T^{2} + 808 T^{3} - 6968 T^{4} + 808 p T^{5} - 29 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.ai_abd_bfc_akia |
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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
−6.22821420968284779875899897002, −5.97780469191070074419715579674, −5.76567886372414996415646447082, −5.39290281425622847450091960082, −5.20561215445966151123828225057, −5.05208187622895771626809352016, −4.91997579049944508920076220854, −4.68134399355515999955952166498, −4.40029525633613950746450632606, −4.27255223110173750643800981371, −4.03289335097330279116193484568, −3.80683882653537407390112024577, −3.52618070267203318116713571954, −3.50371647164698518593554820944, −3.23280847572142610768591003334, −3.08841344103502479183831881320, −2.66643610231533340926665706512, −2.53302049605911130331275087766, −2.38235741896249559606411702123, −2.09194804716568770201147243826, −1.40632054867532144440845520141, −1.00255505817717536108688797525, −0.898599443695549121705973656640, −0.793640741383709337839839067665, −0.12892584956380843569603799464,
0.12892584956380843569603799464, 0.793640741383709337839839067665, 0.898599443695549121705973656640, 1.00255505817717536108688797525, 1.40632054867532144440845520141, 2.09194804716568770201147243826, 2.38235741896249559606411702123, 2.53302049605911130331275087766, 2.66643610231533340926665706512, 3.08841344103502479183831881320, 3.23280847572142610768591003334, 3.50371647164698518593554820944, 3.52618070267203318116713571954, 3.80683882653537407390112024577, 4.03289335097330279116193484568, 4.27255223110173750643800981371, 4.40029525633613950746450632606, 4.68134399355515999955952166498, 4.91997579049944508920076220854, 5.05208187622895771626809352016, 5.20561215445966151123828225057, 5.39290281425622847450091960082, 5.76567886372414996415646447082, 5.97780469191070074419715579674, 6.22821420968284779875899897002
