| L(s) = 1 | − 2·3-s − 3·5-s + 7·7-s − 9-s − 3·11-s − 6·13-s + 6·15-s − 5·17-s − 7·19-s − 14·21-s − 3·23-s + 6·25-s + 7·27-s + 29-s + 10·31-s + 6·33-s − 21·35-s − 2·37-s + 12·39-s − 10·41-s − 12·43-s + 3·45-s + 47-s + 17·49-s + 10·51-s − 10·53-s + 9·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 2.64·7-s − 1/3·9-s − 0.904·11-s − 1.66·13-s + 1.54·15-s − 1.21·17-s − 1.60·19-s − 3.05·21-s − 0.625·23-s + 6/5·25-s + 1.34·27-s + 0.185·29-s + 1.79·31-s + 1.04·33-s − 3.54·35-s − 0.328·37-s + 1.92·39-s − 1.56·41-s − 1.82·43-s + 0.447·45-s + 0.145·47-s + 17/7·49-s + 1.40·51-s − 1.37·53-s + 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 5 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.c_f_f |
| 7 | $S_4\times C_2$ | \( 1 - p T + 32 T^{2} - 102 T^{3} + 32 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.7.ah_bg_ady |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 32 T^{2} + 64 T^{3} + 32 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.d_bg_cm |
| 13 | $S_4\times C_2$ | \( 1 + 6 T + 47 T^{2} + 157 T^{3} + 47 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.g_bv_gb |
| 17 | $S_4\times C_2$ | \( 1 + 5 T + 20 T^{2} + 22 T^{3} + 20 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.f_u_w |
| 19 | $S_4\times C_2$ | \( 1 + 7 T + 46 T^{2} + 160 T^{3} + 46 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.h_bu_ge |
| 29 | $S_4\times C_2$ | \( 1 - T - 3 T^{2} + 90 T^{3} - 3 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ab_ad_dm |
| 31 | $S_4\times C_2$ | \( 1 - 10 T + 107 T^{2} - 567 T^{3} + 107 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ak_ed_avv |
| 37 | $D_{6}$ | \( 1 + 2 T - 17 T^{2} - 364 T^{3} - 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.c_ar_aoa |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 75 T^{2} + 339 T^{3} + 75 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.k_cx_nb |
| 43 | $S_4\times C_2$ | \( 1 + 12 T + 113 T^{2} + 776 T^{3} + 113 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.m_ej_bdw |
| 47 | $S_4\times C_2$ | \( 1 - T + 135 T^{2} - 90 T^{3} + 135 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ab_ff_adm |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 107 T^{2} + 1052 T^{3} + 107 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.k_ed_bom |
| 59 | $S_4\times C_2$ | \( 1 + 10 T + 53 T^{2} - 4 T^{3} + 53 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.k_cb_ae |
| 61 | $S_4\times C_2$ | \( 1 - 13 T + 154 T^{2} - 1372 T^{3} + 154 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.an_fy_acau |
| 67 | $S_4\times C_2$ | \( 1 - 6 T + 41 T^{2} + 380 T^{3} + 41 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ag_bp_oq |
| 71 | $S_4\times C_2$ | \( 1 - 10 T + 113 T^{2} - 545 T^{3} + 113 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ak_ej_auz |
| 73 | $S_4\times C_2$ | \( 1 - 7 T + 185 T^{2} - 786 T^{3} + 185 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ah_hd_abeg |
| 79 | $S_4\times C_2$ | \( 1 - 14 T + 253 T^{2} - 1988 T^{3} + 253 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ao_jt_acym |
| 83 | $S_4\times C_2$ | \( 1 + 16 T + 309 T^{2} + 2688 T^{3} + 309 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.q_lx_dzk |
| 89 | $S_4\times C_2$ | \( 1 + 20 T + 299 T^{2} + 3304 T^{3} + 299 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.u_ln_exc |
| 97 | $S_4\times C_2$ | \( 1 - 5 T + 268 T^{2} - 914 T^{3} + 268 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.af_ki_abje |
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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
−7.40097050022993657081815535172, −7.12697445079586952020779925679, −6.82537170396923712620464884336, −6.66950270243079576230983988145, −6.27440867481884422080990482039, −6.21778572084141571375820239828, −6.15620257490359226609636814054, −5.35766143092606854554730578440, −5.30071932777472081468584372606, −5.15179909896087799003100493505, −4.89611693558258862636353381830, −4.85717947381455206291779302020, −4.61711297626532324867930910286, −4.33325069079043618833223282926, −4.04156069406578933628724374454, −4.00879359994648933836754965444, −3.26417255740592591182168907192, −3.06872533731468032140816288720, −3.04268594789299868524256124234, −2.26391960130301995336771199823, −2.15911809578689873520227617575, −2.12254244134843212810943381046, −1.73471375913162315925989845679, −1.10655266112061041475555851031, −0.891579005693982191098983202280, 0, 0, 0,
0.891579005693982191098983202280, 1.10655266112061041475555851031, 1.73471375913162315925989845679, 2.12254244134843212810943381046, 2.15911809578689873520227617575, 2.26391960130301995336771199823, 3.04268594789299868524256124234, 3.06872533731468032140816288720, 3.26417255740592591182168907192, 4.00879359994648933836754965444, 4.04156069406578933628724374454, 4.33325069079043618833223282926, 4.61711297626532324867930910286, 4.85717947381455206291779302020, 4.89611693558258862636353381830, 5.15179909896087799003100493505, 5.30071932777472081468584372606, 5.35766143092606854554730578440, 6.15620257490359226609636814054, 6.21778572084141571375820239828, 6.27440867481884422080990482039, 6.66950270243079576230983988145, 6.82537170396923712620464884336, 7.12697445079586952020779925679, 7.40097050022993657081815535172
