| L(s) = 1 | + 3·3-s + 3·5-s − 7-s + 6·9-s − 5·11-s − 2·13-s + 9·15-s − 5·19-s − 3·21-s − 11·23-s + 6·25-s + 10·27-s − 4·29-s − 3·31-s − 15·33-s − 3·35-s + 4·37-s − 6·39-s − 16·41-s − 17·43-s + 18·45-s − 14·47-s − 14·49-s + 9·53-s − 15·55-s − 15·57-s − 2·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s − 0.377·7-s + 2·9-s − 1.50·11-s − 0.554·13-s + 2.32·15-s − 1.14·19-s − 0.654·21-s − 2.29·23-s + 6/5·25-s + 1.92·27-s − 0.742·29-s − 0.538·31-s − 2.61·33-s − 0.507·35-s + 0.657·37-s − 0.960·39-s − 2.49·41-s − 2.59·43-s + 2.68·45-s − 2.04·47-s − 2·49-s + 1.23·53-s − 2.02·55-s − 1.98·57-s − 0.260·59-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + T + 15 T^{2} + 16 T^{3} + 15 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.7.b_p_q |
| 11 | $S_4\times C_2$ | \( 1 + 5 T + 3 p T^{2} + 102 T^{3} + 3 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.f_bh_dy |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 21 T^{2} + 8 T^{3} + 21 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.c_v_i |
| 17 | $S_4\times C_2$ | \( 1 + 11 T^{2} + 64 T^{3} + 11 p T^{4} + p^{3} T^{6} \) | 3.17.a_l_cm |
| 19 | $S_4\times C_2$ | \( 1 + 5 T + 3 p T^{2} + 182 T^{3} + 3 p^{2} T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.f_cf_ha |
| 23 | $S_4\times C_2$ | \( 1 + 11 T + 85 T^{2} + 438 T^{3} + 85 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.l_dh_qw |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 73 T^{2} + 240 T^{3} + 73 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.e_cv_jg |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 21 T^{2} + 104 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ae_v_ea |
| 41 | $S_4\times C_2$ | \( 1 + 16 T + 183 T^{2} + 1296 T^{3} + 183 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.q_hb_bxw |
| 43 | $S_4\times C_2$ | \( 1 + 17 T + 153 T^{2} + 998 T^{3} + 153 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.r_fx_bmk |
| 47 | $S_4\times C_2$ | \( 1 + 14 T + 105 T^{2} + 684 T^{3} + 105 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.o_eb_bai |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 111 T^{2} - 722 T^{3} + 111 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aj_eh_abbu |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 127 T^{2} + 336 T^{3} + 127 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.c_ex_my |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 155 T^{2} + 596 T^{3} + 155 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.g_fz_wy |
| 67 | $S_4\times C_2$ | \( 1 + 10 T + 139 T^{2} + 784 T^{3} + 139 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.k_fj_bee |
| 71 | $S_4\times C_2$ | \( 1 - 7 T + 31 T^{2} - 276 T^{3} + 31 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ah_bf_akq |
| 73 | $S_4\times C_2$ | \( 1 - T + 101 T^{2} - 480 T^{3} + 101 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ab_dx_asm |
| 79 | $S_4\times C_2$ | \( 1 + 15 T + 3 p T^{2} + 2086 T^{3} + 3 p^{2} T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.p_jd_dcg |
| 83 | $S_4\times C_2$ | \( 1 + 20 T + 349 T^{2} + 3384 T^{3} + 349 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.u_nl_fae |
| 89 | $S_4\times C_2$ | \( 1 + T + 237 T^{2} + 228 T^{3} + 237 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.89.b_jd_iu |
| 97 | $S_4\times C_2$ | \( 1 - 14 T + 255 T^{2} - 2084 T^{3} + 255 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ao_jv_adce |
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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
−7.53405926607465994951114371456, −6.90100386408861806648171236898, −6.89408179629441774476823954790, −6.82236694788143781969311331479, −6.22753514665919233086574750709, −6.21567835083395500670068353012, −6.20936864838675440043052794422, −5.47934175508132228420146983496, −5.46297143393809925112500459066, −5.27842180390459229372286971986, −4.89497308688444364740706439201, −4.74571768116410202645070113601, −4.47219909456033631981656892803, −4.12851438026065949691804973652, −3.76054776909548256797670099251, −3.72648673423014239458018889441, −3.24213265608310789292719510661, −2.97658184072610919353829171392, −2.94815718158656568387687898624, −2.52541536767232578616575298138, −2.27419938309374310685115211576, −1.98303998014017918278659588070, −1.71375013176384196274179551081, −1.53443961377290238340555922168, −1.35366246390816957199490799969, 0, 0, 0,
1.35366246390816957199490799969, 1.53443961377290238340555922168, 1.71375013176384196274179551081, 1.98303998014017918278659588070, 2.27419938309374310685115211576, 2.52541536767232578616575298138, 2.94815718158656568387687898624, 2.97658184072610919353829171392, 3.24213265608310789292719510661, 3.72648673423014239458018889441, 3.76054776909548256797670099251, 4.12851438026065949691804973652, 4.47219909456033631981656892803, 4.74571768116410202645070113601, 4.89497308688444364740706439201, 5.27842180390459229372286971986, 5.46297143393809925112500459066, 5.47934175508132228420146983496, 6.20936864838675440043052794422, 6.21567835083395500670068353012, 6.22753514665919233086574750709, 6.82236694788143781969311331479, 6.89408179629441774476823954790, 6.90100386408861806648171236898, 7.53405926607465994951114371456
