| L(s) = 1 | − 3·3-s + 3·5-s + 6·9-s − 6·11-s − 9·15-s + 2·17-s − 12·19-s + 2·23-s + 6·25-s − 10·27-s + 4·29-s + 3·31-s + 18·33-s − 6·37-s + 12·41-s − 6·43-s + 18·45-s − 4·47-s − 5·49-s − 6·51-s + 2·53-s − 18·55-s + 36·57-s − 14·59-s + 2·61-s + 2·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s + 2·9-s − 1.80·11-s − 2.32·15-s + 0.485·17-s − 2.75·19-s + 0.417·23-s + 6/5·25-s − 1.92·27-s + 0.742·29-s + 0.538·31-s + 3.13·33-s − 0.986·37-s + 1.87·41-s − 0.914·43-s + 2.68·45-s − 0.583·47-s − 5/7·49-s − 0.840·51-s + 0.274·53-s − 2.42·55-s + 4.76·57-s − 1.82·59-s + 0.256·61-s + 0.244·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.357177450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.357177450\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 31 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} - 18 T^{3} + 5 p T^{4} + p^{3} T^{6} \) | 3.7.a_f_as |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.11.g_bt_fk |
| 13 | $S_4\times C_2$ | \( 1 + 23 T^{2} - 18 T^{3} + 23 p T^{4} + p^{3} T^{6} \) | 3.13.a_x_as |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 11 T^{2} - 24 T^{3} + 11 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ac_l_ay |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) | 3.19.m_eb_ua |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 29 T^{2} - 48 T^{3} + 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ac_bd_abw |
| 29 | $S_4\times C_2$ | \( 1 - 4 T + 45 T^{2} - 286 T^{3} + 45 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ae_bt_ala |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 107 T^{2} + 402 T^{3} + 107 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.g_ed_pm |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) | 3.41.am_gp_aboi |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.43.g_fl_ue |
| 47 | $S_4\times C_2$ | \( 1 + 4 T + 61 T^{2} + 244 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.e_cj_jk |
| 53 | $S_4\times C_2$ | \( 1 - 2 T + 75 T^{2} - 176 T^{3} + 75 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ac_cx_agu |
| 59 | $S_4\times C_2$ | \( 1 + 14 T + 195 T^{2} + 1418 T^{3} + 195 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.o_hn_cco |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 19 T^{2} - 268 T^{3} + 19 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ac_t_aki |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + T^{2} + 554 T^{3} + p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ac_b_vi |
| 71 | $S_4\times C_2$ | \( 1 - 4 T + 63 T^{2} + 134 T^{3} + 63 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ae_cl_fe |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 87 T^{2} - 1082 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ag_dj_abpq |
| 79 | $S_4\times C_2$ | \( 1 + 4 T + 117 T^{2} + 20 T^{3} + 117 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.e_en_u |
| 83 | $S_4\times C_2$ | \( 1 - 10 T + 241 T^{2} - 1576 T^{3} + 241 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ak_jh_aciq |
| 89 | $S_4\times C_2$ | \( 1 - 34 T + 585 T^{2} - 6538 T^{3} + 585 p T^{4} - 34 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.abi_wn_ajrm |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 147 T^{2} - 1264 T^{3} + 147 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ai_fr_abwq |
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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
−6.76790176777894508877390193061, −6.64107706109629690496677995046, −6.38348145431119371657913563509, −6.29842229260020471658399797930, −5.92092849123070117874872365641, −5.86592016692915793465715288050, −5.77983408429290111430625420143, −5.16452024356310208219014001132, −5.12566136270418499630087410249, −4.99622388683140814992154213904, −4.54633828090479290675596983379, −4.52022567836723457981312935291, −4.48050853624095071993639395543, −3.73918691858171651161531748168, −3.59199644764385345733844506946, −3.30880918631133960698844545108, −2.89217084999232248716377481333, −2.51586661301099954946807096985, −2.45930143481227438312705851916, −1.92397151723001513323494834672, −1.73439341764415316708003545926, −1.70140029846707046849484947243, −0.864253499474762383485478138928, −0.51910176755652387303574761575, −0.47222556087413309400018602582,
0.47222556087413309400018602582, 0.51910176755652387303574761575, 0.864253499474762383485478138928, 1.70140029846707046849484947243, 1.73439341764415316708003545926, 1.92397151723001513323494834672, 2.45930143481227438312705851916, 2.51586661301099954946807096985, 2.89217084999232248716377481333, 3.30880918631133960698844545108, 3.59199644764385345733844506946, 3.73918691858171651161531748168, 4.48050853624095071993639395543, 4.52022567836723457981312935291, 4.54633828090479290675596983379, 4.99622388683140814992154213904, 5.12566136270418499630087410249, 5.16452024356310208219014001132, 5.77983408429290111430625420143, 5.86592016692915793465715288050, 5.92092849123070117874872365641, 6.29842229260020471658399797930, 6.38348145431119371657913563509, 6.64107706109629690496677995046, 6.76790176777894508877390193061
