| L(s) = 1 | + 4·5-s − 11·11-s − 2·13-s + 3·17-s + 2·19-s − 10·23-s − 25-s + 3·29-s + 11·31-s + 3·37-s + 3·41-s − 5·43-s − 3·47-s − 17·49-s − 44·55-s − 4·59-s − 61-s − 8·65-s − 4·67-s − 29·71-s − 17·73-s + 4·79-s + 18·83-s + 12·85-s − 6·89-s + 8·95-s − 24·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 3.31·11-s − 0.554·13-s + 0.727·17-s + 0.458·19-s − 2.08·23-s − 1/5·25-s + 0.557·29-s + 1.97·31-s + 0.493·37-s + 0.468·41-s − 0.762·43-s − 0.437·47-s − 2.42·49-s − 5.93·55-s − 0.520·59-s − 0.128·61-s − 0.992·65-s − 0.488·67-s − 3.44·71-s − 1.98·73-s + 0.450·79-s + 1.97·83-s + 1.30·85-s − 0.635·89-s + 0.820·95-s − 2.43·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 41 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 5 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 38 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ae_r_abm |
| 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_c |
| 11 | $S_4\times C_2$ | \( 1 + p T + 70 T^{2} + 279 T^{3} + 70 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) | 3.11.l_cs_kt |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 27 T^{2} + 62 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.c_bb_ck |
| 17 | $S_4\times C_2$ | \( 1 - 3 T + 18 T^{2} - 121 T^{3} + 18 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ad_s_aer |
| 19 | $S_4\times C_2$ | \( 1 - 2 T - 7 T^{2} + 102 T^{3} - 7 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ac_ah_dy |
| 23 | $S_4\times C_2$ | \( 1 + 10 T + 79 T^{2} + 402 T^{3} + 79 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.k_db_pm |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 20 T^{2} - 19 T^{3} + 20 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ad_u_at |
| 31 | $S_4\times C_2$ | \( 1 - 11 T + 88 T^{2} - 575 T^{3} + 88 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.al_dk_awd |
| 37 | $S_4\times C_2$ | \( 1 - 3 T + 66 T^{2} - 195 T^{3} + 66 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ad_co_ahn |
| 43 | $S_4\times C_2$ | \( 1 + 5 T + 100 T^{2} + 293 T^{3} + 100 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.f_dw_lh |
| 47 | $S_4\times C_2$ | \( 1 + 3 T + 108 T^{2} + 193 T^{3} + 108 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.d_ee_hl |
| 53 | $S_4\times C_2$ | \( 1 + 119 T^{2} - 76 T^{3} + 119 p T^{4} + p^{3} T^{6} \) | 3.53.a_ep_acy |
| 59 | $S_4\times C_2$ | \( 1 + 4 T + 89 T^{2} + 488 T^{3} + 89 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.e_dl_su |
| 61 | $S_4\times C_2$ | \( 1 + T + 98 T^{2} - 195 T^{3} + 98 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.61.b_du_ahn |
| 67 | $S_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 376 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.e_z_om |
| 71 | $S_4\times C_2$ | \( 1 + 29 T + 456 T^{2} + 4611 T^{3} + 456 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.bd_ro_gvj |
| 73 | $S_4\times C_2$ | \( 1 + 17 T + 294 T^{2} + 2549 T^{3} + 294 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.r_li_dub |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 157 T^{2} - 232 T^{3} + 157 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ae_gb_aiy |
| 83 | $S_4\times C_2$ | \( 1 - 18 T + 287 T^{2} - 2698 T^{3} + 287 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.as_lb_adzu |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 215 T^{2} + 820 T^{3} + 215 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.g_ih_bfo |
| 97 | $S_4\times C_2$ | \( 1 + 24 T + 383 T^{2} + 4118 T^{3} + 383 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.y_ot_gck |
| 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
−7.72004277029739007276408620535, −7.19702000027712724533752206293, −7.16835095194452544154353714664, −6.79853733224971367510752424950, −6.32952291800851219842389121461, −6.25683266030196971399123083879, −6.09514651191283306835197127630, −5.67929012836439786473160886006, −5.57895696175660849258877473432, −5.55150488716912896994645493217, −5.21322396713396308724327899024, −4.77114765114143575168592406104, −4.75652659340300641133111845076, −4.39972653480907626237721716553, −4.26745246511952447143370731154, −3.70490199038905121611333106140, −3.39177890184136063600991953437, −3.04636973115548647227037871027, −2.84143565278835135905710249789, −2.46369891517807967490325770169, −2.37236848140754011962657673040, −2.34807411504052347336923384129, −1.56126685567654665656631777956, −1.37051705678792226670976185230, −1.31894565269408976972288692140, 0, 0, 0,
1.31894565269408976972288692140, 1.37051705678792226670976185230, 1.56126685567654665656631777956, 2.34807411504052347336923384129, 2.37236848140754011962657673040, 2.46369891517807967490325770169, 2.84143565278835135905710249789, 3.04636973115548647227037871027, 3.39177890184136063600991953437, 3.70490199038905121611333106140, 4.26745246511952447143370731154, 4.39972653480907626237721716553, 4.75652659340300641133111845076, 4.77114765114143575168592406104, 5.21322396713396308724327899024, 5.55150488716912896994645493217, 5.57895696175660849258877473432, 5.67929012836439786473160886006, 6.09514651191283306835197127630, 6.25683266030196971399123083879, 6.32952291800851219842389121461, 6.79853733224971367510752424950, 7.16835095194452544154353714664, 7.19702000027712724533752206293, 7.72004277029739007276408620535
