| L(s) = 1 | + 3·3-s + 3·5-s + 4·7-s + 4·9-s − 2·11-s + 5·13-s + 9·15-s − 3·17-s − 19-s + 12·21-s − 4·23-s + 6·25-s + 5·27-s + 7·29-s − 3·31-s − 6·33-s + 12·35-s + 12·37-s + 15·39-s + 10·41-s + 6·43-s + 12·45-s − 5·47-s + 49-s − 9·51-s − 13·53-s − 6·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s + 1.51·7-s + 4/3·9-s − 0.603·11-s + 1.38·13-s + 2.32·15-s − 0.727·17-s − 0.229·19-s + 2.61·21-s − 0.834·23-s + 6/5·25-s + 0.962·27-s + 1.29·29-s − 0.538·31-s − 1.04·33-s + 2.02·35-s + 1.97·37-s + 2.40·39-s + 1.56·41-s + 0.914·43-s + 1.78·45-s − 0.729·47-s + 1/7·49-s − 1.26·51-s − 1.78·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 17^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.019259566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.019259566\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 17 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 - p T + 5 T^{2} - 8 T^{3} + 5 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.3.ad_f_ai |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 15 T^{2} - 52 T^{3} + 15 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_p_aca |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 23 T^{2} + 28 T^{3} + 23 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.c_x_bc |
| 13 | $S_4\times C_2$ | \( 1 - 5 T + 31 T^{2} - 98 T^{3} + 31 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.af_bf_adu |
| 19 | $S_4\times C_2$ | \( 1 + T + 37 T^{2} - 2 T^{3} + 37 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.19.b_bl_ac |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 63 T^{2} + 180 T^{3} + 63 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.e_cl_gy |
| 29 | $S_4\times C_2$ | \( 1 - 7 T + 83 T^{2} - 338 T^{3} + 83 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ah_df_ana |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 16 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.d_bh_q |
| 37 | $S_4\times C_2$ | \( 1 - 12 T + 131 T^{2} - 872 T^{3} + 131 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.am_fb_abho |
| 41 | $S_4\times C_2$ | \( 1 - 10 T + 111 T^{2} - 636 T^{3} + 111 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.ak_eh_aym |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.43.ag_fl_aue |
| 47 | $S_4\times C_2$ | \( 1 + 5 T + 129 T^{2} + 474 T^{3} + 129 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.f_ez_sg |
| 53 | $S_4\times C_2$ | \( 1 + 13 T + 39 T^{2} - 186 T^{3} + 39 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.n_bn_ahe |
| 59 | $S_4\times C_2$ | \( 1 + 21 T + 221 T^{2} + 1702 T^{3} + 221 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.v_in_cnm |
| 61 | $S_4\times C_2$ | \( 1 - 13 T + 223 T^{2} - 1606 T^{3} + 223 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.an_ip_acju |
| 67 | $S_4\times C_2$ | \( 1 - 14 T + 93 T^{2} - 428 T^{3} + 93 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ao_dp_aqm |
| 71 | $S_4\times C_2$ | \( 1 + 11 T + 157 T^{2} + 1152 T^{3} + 157 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.l_gb_bsi |
| 73 | $S_4\times C_2$ | \( 1 + 5 T + 207 T^{2} + 734 T^{3} + 207 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.f_hz_bcg |
| 79 | $S_4\times C_2$ | \( 1 + 18 T + 243 T^{2} + 2684 T^{3} + 243 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.s_jj_dzg |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 189 T^{2} - 504 T^{3} + 189 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ae_hh_atk |
| 89 | $S_4\times C_2$ | \( 1 - 11 T + 111 T^{2} - 1422 T^{3} + 111 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.al_eh_accs |
| 97 | $S_4\times C_2$ | \( 1 - 37 T + 731 T^{2} - 8862 T^{3} + 731 p T^{4} - 37 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.abl_bcd_ancw |
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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
−9.371808152746956672754849208271, −8.968770927909887985323401336851, −8.704829752961983824026085144789, −8.575465411837267170125146007826, −8.161686342034143162319410313600, −7.969288822287353779997446574197, −7.83885326553108985032820717044, −7.38394132900507575432246260437, −7.18100612939431627095540236842, −6.50115932770953870182169118787, −6.13551854340561028511955536143, −6.13119314931470421225192530411, −5.96227652900070639708923291472, −5.16786907100812694942599915210, −4.86178435168030921503846920255, −4.80998595787348960503927741885, −4.18526793105726993359624364294, −4.01390126118031892267127499418, −3.50029186207369379706280188496, −2.88395302779758627262914186463, −2.64511267181559028188249240811, −2.48002525207334108846198272439, −1.71637837809531664912056149507, −1.66014867404812180991374356439, −0.991895880073720539207942367785,
0.991895880073720539207942367785, 1.66014867404812180991374356439, 1.71637837809531664912056149507, 2.48002525207334108846198272439, 2.64511267181559028188249240811, 2.88395302779758627262914186463, 3.50029186207369379706280188496, 4.01390126118031892267127499418, 4.18526793105726993359624364294, 4.80998595787348960503927741885, 4.86178435168030921503846920255, 5.16786907100812694942599915210, 5.96227652900070639708923291472, 6.13119314931470421225192530411, 6.13551854340561028511955536143, 6.50115932770953870182169118787, 7.18100612939431627095540236842, 7.38394132900507575432246260437, 7.83885326553108985032820717044, 7.969288822287353779997446574197, 8.161686342034143162319410313600, 8.575465411837267170125146007826, 8.704829752961983824026085144789, 8.968770927909887985323401336851, 9.371808152746956672754849208271
