| L(s) = 1 | − 3·2-s + 3-s + 6·4-s + 3·5-s − 3·6-s − 5·7-s − 10·8-s − 6·9-s − 9·10-s − 4·11-s + 6·12-s + 15·14-s + 3·15-s + 15·16-s + 10·17-s + 18·18-s − 8·19-s + 18·20-s − 5·21-s + 12·22-s − 3·23-s − 10·24-s + 6·25-s − 8·27-s − 30·28-s + 3·29-s − 9·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.577·3-s + 3·4-s + 1.34·5-s − 1.22·6-s − 1.88·7-s − 3.53·8-s − 2·9-s − 2.84·10-s − 1.20·11-s + 1.73·12-s + 4.00·14-s + 0.774·15-s + 15/4·16-s + 2.42·17-s + 4.24·18-s − 1.83·19-s + 4.02·20-s − 1.09·21-s + 2.55·22-s − 0.625·23-s − 2.04·24-s + 6/5·25-s − 1.53·27-s − 5.66·28-s + 0.557·29-s − 1.64·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 13 | | \( 1 \) | |
| good | 3 | $A_4\times C_2$ | \( 1 - T + 7 T^{2} - 5 T^{3} + 7 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ab_h_af |
| 7 | $A_4\times C_2$ | \( 1 + 5 T + 27 T^{2} + 71 T^{3} + 27 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.f_bb_ct |
| 11 | $A_4\times C_2$ | \( 1 + 4 T + 29 T^{2} + 80 T^{3} + 29 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.e_bd_dc |
| 17 | $A_4\times C_2$ | \( 1 - 10 T + 75 T^{2} - 348 T^{3} + 75 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ak_cx_ank |
| 19 | $A_4\times C_2$ | \( 1 + 8 T + 69 T^{2} + 296 T^{3} + 69 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.i_cr_lk |
| 23 | $A_4\times C_2$ | \( 1 + 3 T + p T^{2} - T^{3} + p^{2} T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.d_x_ab |
| 29 | $A_4\times C_2$ | \( 1 - 3 T + 83 T^{2} - 175 T^{3} + 83 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ad_df_agt |
| 31 | $A_4\times C_2$ | \( 1 + 8 T + 49 T^{2} + 152 T^{3} + 49 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.i_bx_fw |
| 37 | $A_4\times C_2$ | \( 1 + 8 T + 95 T^{2} + 528 T^{3} + 95 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.i_dr_ui |
| 41 | $A_4\times C_2$ | \( 1 + 11 T + 49 T^{2} + 75 T^{3} + 49 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.l_bx_cx |
| 43 | $A_4\times C_2$ | \( 1 + T + 99 T^{2} + p T^{3} + 99 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.43.b_dv_br |
| 47 | $A_4\times C_2$ | \( 1 + 15 T + 167 T^{2} + 1339 T^{3} + 167 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.p_gl_bzn |
| 53 | $A_4\times C_2$ | \( 1 - 4 T + 43 T^{2} + 144 T^{3} + 43 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ae_br_fo |
| 59 | $A_4\times C_2$ | \( 1 + 24 T + 341 T^{2} + 3176 T^{3} + 341 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.y_nd_ese |
| 61 | $A_4\times C_2$ | \( 1 + 15 T + 167 T^{2} + 1297 T^{3} + 167 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.p_gl_bxx |
| 67 | $A_4\times C_2$ | \( 1 + 5 T + 123 T^{2} + 839 T^{3} + 123 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.f_et_bgh |
| 71 | $A_4\times C_2$ | \( 1 + 32 T + 545 T^{2} + 5656 T^{3} + 545 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.bg_uz_ijo |
| 73 | $A_4\times C_2$ | \( 1 - 26 T + 379 T^{2} - 3692 T^{3} + 379 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aba_op_afma |
| 79 | $A_4\times C_2$ | \( 1 - 2 T + 117 T^{2} + 28 T^{3} + 117 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ac_en_bc |
| 83 | $A_4\times C_2$ | \( 1 + 7 T + 95 T^{2} + 875 T^{3} + 95 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.h_dr_bhr |
| 89 | $A_4\times C_2$ | \( 1 - 17 T + 137 T^{2} - 829 T^{3} + 137 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ar_fh_abfx |
| 97 | $A_4\times C_2$ | \( 1 - 6 T + 107 T^{2} - 52 T^{3} + 107 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ag_ed_aca |
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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
−8.951957386054073538911169039387, −8.388003087986880720697198289981, −8.176328202523811002819318548305, −8.155932114934338152166094577584, −7.77574267136148670660816568122, −7.45005810250461252468519354836, −7.38564040461661719535119942861, −6.62005487638181958211139928004, −6.51535740340499805263126404481, −6.48926621105491025992963571849, −6.02138033230657013565327411152, −5.93891504604407512413047824106, −5.68479847413728761935270105152, −5.25604248216279024062638891817, −4.95423418070632430006890707802, −4.80861270103080427272032062753, −3.74693065610866476837550067526, −3.46436727402739108592896516717, −3.38297194691024270094409252993, −2.94742121664306434268952366890, −2.67509984951510738860333429198, −2.66874411405003533889139912435, −1.92798746696859126923452526842, −1.59241437633239548266475978420, −1.43381956222028651570986563363, 0, 0, 0,
1.43381956222028651570986563363, 1.59241437633239548266475978420, 1.92798746696859126923452526842, 2.66874411405003533889139912435, 2.67509984951510738860333429198, 2.94742121664306434268952366890, 3.38297194691024270094409252993, 3.46436727402739108592896516717, 3.74693065610866476837550067526, 4.80861270103080427272032062753, 4.95423418070632430006890707802, 5.25604248216279024062638891817, 5.68479847413728761935270105152, 5.93891504604407512413047824106, 6.02138033230657013565327411152, 6.48926621105491025992963571849, 6.51535740340499805263126404481, 6.62005487638181958211139928004, 7.38564040461661719535119942861, 7.45005810250461252468519354836, 7.77574267136148670660816568122, 8.155932114934338152166094577584, 8.176328202523811002819318548305, 8.388003087986880720697198289981, 8.951957386054073538911169039387
