| L(s) = 1 | + 2-s + 2·3-s − 4-s − 2·5-s + 2·6-s − 8-s + 9-s − 2·10-s + 2·11-s − 2·12-s − 3·13-s − 4·15-s − 16-s − 4·17-s + 18-s + 4·19-s + 2·20-s + 2·22-s + 10·23-s − 2·24-s − 8·25-s − 3·26-s + 24·29-s − 4·30-s + 4·31-s − 32-s + 4·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s − 1/2·4-s − 0.894·5-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.603·11-s − 0.577·12-s − 0.832·13-s − 1.03·15-s − 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.426·22-s + 2.08·23-s − 0.408·24-s − 8/5·25-s − 0.588·26-s + 4.45·29-s − 0.730·30-s + 0.718·31-s − 0.176·32-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.704908367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.704908367\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.2.ab_c_ac |
| 3 | $S_4\times C_2$ | \( 1 - 2 T + p T^{2} - 4 T^{3} + p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ac_d_ae |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 12 T^{2} + 18 T^{3} + 12 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.c_m_s |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 27 T^{2} - 36 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ac_bb_abk |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 140 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.e_bp_fk |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 58 T^{2} - 148 T^{3} + 58 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ae_cg_afs |
| 23 | $S_4\times C_2$ | \( 1 - 10 T + 70 T^{2} - 324 T^{3} + 70 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ak_cs_amm |
| 29 | $S_4\times C_2$ | \( 1 - 24 T + 272 T^{2} - 1846 T^{3} + 272 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ay_km_acta |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 74 T^{2} - 264 T^{3} + 74 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ae_cw_ake |
| 37 | $S_4\times C_2$ | \( 1 + 53 T^{2} - 124 T^{3} + 53 p T^{4} + p^{3} T^{6} \) | 3.37.a_cb_aeu |
| 41 | $S_4\times C_2$ | \( 1 + 2 T + 95 T^{2} + 172 T^{3} + 95 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.c_dr_gq |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 58 T^{2} - 232 T^{3} + 58 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ak_cg_aiy |
| 47 | $S_4\times C_2$ | \( 1 - 8 T + 62 T^{2} - 208 T^{3} + 62 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ai_ck_aia |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 124 T^{2} - 870 T^{3} + 124 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ai_eu_abhm |
| 59 | $S_4\times C_2$ | \( 1 - 4 T + 21 T^{2} + 216 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ae_v_ii |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.61.ag_hn_abcm |
| 67 | $S_4\times C_2$ | \( 1 + 12 T + 77 T^{2} + 632 T^{3} + 77 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.m_cz_yi |
| 71 | $S_4\times C_2$ | \( 1 + 6 T + 191 T^{2} + 868 T^{3} + 191 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.g_hj_bhk |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 120 T^{2} - 1186 T^{3} + 120 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ak_eq_abtq |
| 79 | $S_4\times C_2$ | \( 1 + 14 T + 242 T^{2} + 2196 T^{3} + 242 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.o_ji_dgm |
| 83 | $S_4\times C_2$ | \( 1 - 12 T - 22 T^{2} + 1276 T^{3} - 22 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.am_aw_bxc |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 172 T^{2} - 66 T^{3} + 172 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.c_gq_aco |
| 97 | $S_4\times C_2$ | \( 1 - 10 T + 320 T^{2} - 1962 T^{3} + 320 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ak_mi_acxm |
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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
−9.341630560827381252799713591963, −8.873697915208596104311527575212, −8.858273253139612343567181957335, −8.751367970145830128439756053077, −8.262540967048320716968127345186, −7.936017168618540638475479885292, −7.78625288877059975270319363518, −7.38550888749104813822095304232, −6.94361741193310573900581815702, −6.81802789260794889648396775126, −6.59010937519673736239124424302, −5.97417129810271293235761021520, −5.76250169193704556090730717133, −5.13183137493496915966746099193, −4.94775821718714729055606283018, −4.55818398130437604350524277279, −4.36088514355770213159537729886, −4.02691821596254686839092015711, −3.77491097616630385233766186485, −3.00857864923386076986054653525, −2.91868243165336358882903193607, −2.64983576598868699160395521306, −2.13648582623965500768110420761, −1.16643628535130692932801266828, −0.71272129291482955258320604293,
0.71272129291482955258320604293, 1.16643628535130692932801266828, 2.13648582623965500768110420761, 2.64983576598868699160395521306, 2.91868243165336358882903193607, 3.00857864923386076986054653525, 3.77491097616630385233766186485, 4.02691821596254686839092015711, 4.36088514355770213159537729886, 4.55818398130437604350524277279, 4.94775821718714729055606283018, 5.13183137493496915966746099193, 5.76250169193704556090730717133, 5.97417129810271293235761021520, 6.59010937519673736239124424302, 6.81802789260794889648396775126, 6.94361741193310573900581815702, 7.38550888749104813822095304232, 7.78625288877059975270319363518, 7.936017168618540638475479885292, 8.262540967048320716968127345186, 8.751367970145830128439756053077, 8.858273253139612343567181957335, 8.873697915208596104311527575212, 9.341630560827381252799713591963
