| L(s) = 1 | + 3·2-s + 4·4-s − 2·7-s + 4·8-s − 2·11-s + 6·13-s − 6·14-s + 3·16-s + 4·17-s − 8·19-s − 6·22-s + 14·23-s + 18·26-s − 8·28-s − 16·29-s − 3·31-s − 32-s + 12·34-s + 8·37-s − 24·38-s − 4·41-s + 2·43-s − 8·44-s + 42·46-s + 14·47-s − 5·49-s + 24·52-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 2·4-s − 0.755·7-s + 1.41·8-s − 0.603·11-s + 1.66·13-s − 1.60·14-s + 3/4·16-s + 0.970·17-s − 1.83·19-s − 1.27·22-s + 2.91·23-s + 3.53·26-s − 1.51·28-s − 2.97·29-s − 0.538·31-s − 0.176·32-s + 2.05·34-s + 1.31·37-s − 3.89·38-s − 0.624·41-s + 0.304·43-s − 1.20·44-s + 6.19·46-s + 2.04·47-s − 5/7·49-s + 3.32·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.28328903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.28328903\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 - 3 T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.2.ad_f_ah |
| 7 | $S_4\times C_2$ | \( 1 + 2 T + 9 T^{2} + 38 T^{3} + 9 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.c_j_bm |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 21 T^{2} + 36 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.c_v_bk |
| 13 | $S_4\times C_2$ | \( 1 - 6 T + 47 T^{2} - 154 T^{3} + 47 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ag_bv_afy |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 47 T^{2} - 116 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_bv_aem |
| 19 | $S_4\times C_2$ | \( 1 + 8 T + 3 p T^{2} + 272 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.i_cf_km |
| 23 | $S_4\times C_2$ | \( 1 - 14 T + 129 T^{2} - 720 T^{3} + 129 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ao_ez_abbs |
| 29 | $S_4\times C_2$ | \( 1 + 16 T + 149 T^{2} + 938 T^{3} + 149 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.q_ft_bkc |
| 37 | $S_4\times C_2$ | \( 1 - 8 T + 3 p T^{2} - 14 p T^{3} + 3 p^{2} T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ai_eh_aty |
| 41 | $S_4\times C_2$ | \( 1 + 4 T + 35 T^{2} + 344 T^{3} + 35 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.e_bj_ng |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 69 T^{2} + 28 T^{3} + 69 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ac_cr_bc |
| 47 | $S_4\times C_2$ | \( 1 - 14 T + 49 T^{2} + 72 T^{3} + 49 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ao_bx_cu |
| 53 | $S_4\times C_2$ | \( 1 - 8 T + 87 T^{2} - 948 T^{3} + 87 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ai_dj_abkm |
| 59 | $S_4\times C_2$ | \( 1 - 26 T + 379 T^{2} - 3534 T^{3} + 379 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.aba_op_affy |
| 61 | $S_4\times C_2$ | \( 1 + 18 T + 227 T^{2} + 1900 T^{3} + 227 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.s_it_cvc |
| 67 | $S_4\times C_2$ | \( 1 + 65 T^{2} + 274 T^{3} + 65 p T^{4} + p^{3} T^{6} \) | 3.67.a_cn_ko |
| 71 | $S_4\times C_2$ | \( 1 + 4 T + 215 T^{2} + 566 T^{3} + 215 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.e_ih_vu |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 155 T^{2} - 1618 T^{3} + 155 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.am_fz_ackg |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 93 T^{2} - 1132 T^{3} + 93 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ae_dp_abro |
| 83 | $S_4\times C_2$ | \( 1 + 10 T + 205 T^{2} + 1272 T^{3} + 205 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.k_hx_bwy |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 249 T^{2} - 1018 T^{3} + 249 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ag_jp_abne |
| 97 | $S_4\times C_2$ | \( 1 + 8 T + 51 T^{2} - 160 T^{3} + 51 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.i_bz_age |
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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
−6.90109924633652150575161854755, −6.89297254607190134576563131414, −6.39435333405375968698295999487, −6.31922024125828311458193441963, −5.83196598435713284189776575829, −5.75676985676852443743269371214, −5.60020528458239895231990760703, −5.28033980189679140955587911118, −5.19390875423199092104540814304, −4.98828187865129943207759011858, −4.36365037931701051523788501655, −4.35829330284935820124749644052, −4.22729765559564952286584631147, −3.73728238165997562793274769692, −3.63180596333575008482695194413, −3.60969361080035873793130170276, −3.05954976478064808157407737082, −2.98924334114084048289723725995, −2.54120793065312360518552428452, −2.28092561004028772656574322982, −2.04102122887970288031289041428, −1.50229176173118707561887495466, −1.29810189309947947989595162529, −0.70609500695673463894405471007, −0.44177312916424052158314077945,
0.44177312916424052158314077945, 0.70609500695673463894405471007, 1.29810189309947947989595162529, 1.50229176173118707561887495466, 2.04102122887970288031289041428, 2.28092561004028772656574322982, 2.54120793065312360518552428452, 2.98924334114084048289723725995, 3.05954976478064808157407737082, 3.60969361080035873793130170276, 3.63180596333575008482695194413, 3.73728238165997562793274769692, 4.22729765559564952286584631147, 4.35829330284935820124749644052, 4.36365037931701051523788501655, 4.98828187865129943207759011858, 5.19390875423199092104540814304, 5.28033980189679140955587911118, 5.60020528458239895231990760703, 5.75676985676852443743269371214, 5.83196598435713284189776575829, 6.31922024125828311458193441963, 6.39435333405375968698295999487, 6.89297254607190134576563131414, 6.90109924633652150575161854755
