| L(s) = 1 | + 3·2-s − 2·3-s + 6·4-s − 2·5-s − 6·6-s − 3·7-s + 10·8-s − 9-s − 6·10-s + 4·11-s − 12·12-s − 9·14-s + 4·15-s + 15·16-s − 4·17-s − 3·18-s − 6·19-s − 12·20-s + 6·21-s + 12·22-s − 20·24-s − 3·25-s + 4·27-s − 18·28-s + 2·29-s + 12·30-s + 8·31-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 1.15·3-s + 3·4-s − 0.894·5-s − 2.44·6-s − 1.13·7-s + 3.53·8-s − 1/3·9-s − 1.89·10-s + 1.20·11-s − 3.46·12-s − 2.40·14-s + 1.03·15-s + 15/4·16-s − 0.970·17-s − 0.707·18-s − 1.37·19-s − 2.68·20-s + 1.30·21-s + 2.55·22-s − 4.08·24-s − 3/5·25-s + 0.769·27-s − 3.40·28-s + 0.371·29-s + 2.19·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 7^{3} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 23 | | \( 1 \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 8 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.c_f_i |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 24 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.c_h_y |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 17 T^{2} - 56 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ae_r_ace |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.13.a_bn_a |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 3 p T^{2} + 132 T^{3} + 3 p^{2} T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.e_bz_fc |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 53 T^{2} + 220 T^{3} + 53 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.g_cb_im |
| 29 | $S_4\times C_2$ | \( 1 - 2 T + 75 T^{2} - 108 T^{3} + 75 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ac_cx_aee |
| 31 | $S_4\times C_2$ | \( 1 - 8 T + 69 T^{2} - 404 T^{3} + 69 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ai_cr_apo |
| 37 | $S_4\times C_2$ | \( 1 + 8 T + 119 T^{2} + 576 T^{3} + 119 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.i_ep_we |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 71 T^{2} + 500 T^{3} + 71 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.g_ct_tg |
| 43 | $S_4\times C_2$ | \( 1 + 16 T + 3 p T^{2} + 784 T^{3} + 3 p^{2} T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.q_ez_bee |
| 47 | $S_4\times C_2$ | \( 1 + 20 T + 269 T^{2} + 2140 T^{3} + 269 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.u_kj_dei |
| 53 | $S_4\times C_2$ | \( 1 - 16 T + 231 T^{2} - 1776 T^{3} + 231 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aq_ix_acqi |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 29 T^{2} - 560 T^{3} + 29 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.c_bd_avo |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 31 T^{2} - 208 T^{3} + 31 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.c_bf_aia |
| 67 | $S_4\times C_2$ | \( 1 + 8 T + 89 T^{2} + 144 T^{3} + 89 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.i_dl_fo |
| 71 | $S_4\times C_2$ | \( 1 - 16 T + 277 T^{2} - 2304 T^{3} + 277 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.aq_kr_adkq |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 103 T^{2} - 364 T^{3} + 103 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ak_dz_aoa |
| 79 | $S_4\times C_2$ | \( 1 + 32 T + 525 T^{2} + 5552 T^{3} + 525 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.bg_uf_ifo |
| 83 | $S_4\times C_2$ | \( 1 + 18 T + 341 T^{2} + 3092 T^{3} + 341 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.s_nd_eoy |
| 89 | $S_4\times C_2$ | \( 1 + 4 T - 37 T^{2} - 1468 T^{3} - 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.e_abl_acem |
| 97 | $S_4\times C_2$ | \( 1 + 4 T + 219 T^{2} + 508 T^{3} + 219 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.e_il_to |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.91099258495700703863249100261, −6.87168376685624287740164641486, −6.79602597328754957427958792715, −6.62599483329784069847766452048, −6.22781699838624360307673693818, −6.15451337295205932159569737080, −6.09670040971913850898397901928, −5.47023382766370350104990251593, −5.43849729164993969347935842904, −5.36510894978694495486009974349, −4.90804057746184826747976722695, −4.69527737822670485613752413904, −4.41594487249003409416121979917, −4.17662208658433738358544195406, −4.07221040239929000023867036820, −3.79321759951337215455505897231, −3.48439811852134867917411840006, −3.20324584360657364824676145153, −3.09100397201710258792405945561, −2.71375704849499150873436867012, −2.47039032488723462516520679090, −1.99137839410306399288828194839, −1.77304306612733231823592388545, −1.37035672813975733090912511742, −1.04687741574210022200688877935, 0, 0, 0,
1.04687741574210022200688877935, 1.37035672813975733090912511742, 1.77304306612733231823592388545, 1.99137839410306399288828194839, 2.47039032488723462516520679090, 2.71375704849499150873436867012, 3.09100397201710258792405945561, 3.20324584360657364824676145153, 3.48439811852134867917411840006, 3.79321759951337215455505897231, 4.07221040239929000023867036820, 4.17662208658433738358544195406, 4.41594487249003409416121979917, 4.69527737822670485613752413904, 4.90804057746184826747976722695, 5.36510894978694495486009974349, 5.43849729164993969347935842904, 5.47023382766370350104990251593, 6.09670040971913850898397901928, 6.15451337295205932159569737080, 6.22781699838624360307673693818, 6.62599483329784069847766452048, 6.79602597328754957427958792715, 6.87168376685624287740164641486, 6.91099258495700703863249100261
