| L(s) = 1 | − 3·2-s − 2·3-s + 6·4-s + 5·5-s + 6·6-s + 7-s − 10·8-s − 4·9-s − 15·10-s − 3·11-s − 12·12-s + 2·13-s − 3·14-s − 10·15-s + 15·16-s − 5·17-s + 12·18-s − 10·19-s + 30·20-s − 2·21-s + 9·22-s + 20·24-s + 4·25-s − 6·26-s + 13·27-s + 6·28-s + 30·30-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 1.15·3-s + 3·4-s + 2.23·5-s + 2.44·6-s + 0.377·7-s − 3.53·8-s − 4/3·9-s − 4.74·10-s − 0.904·11-s − 3.46·12-s + 0.554·13-s − 0.801·14-s − 2.58·15-s + 15/4·16-s − 1.21·17-s + 2.82·18-s − 2.29·19-s + 6.70·20-s − 0.436·21-s + 1.91·22-s + 4.08·24-s + 4/5·25-s − 1.17·26-s + 2.50·27-s + 1.13·28-s + 5.47·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 29 | | \( 1 \) | |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 T + 8 T^{2} + 11 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.c_i_l |
| 5 | $A_4\times C_2$ | \( 1 - p T + 21 T^{2} - 51 T^{3} + 21 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.5.af_v_abz |
| 7 | $A_4\times C_2$ | \( 1 - T + 12 T^{2} - 13 T^{3} + 12 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ab_m_an |
| 11 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} + 39 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.d_p_bn |
| 13 | $A_4\times C_2$ | \( 1 - 2 T + 24 T^{2} - 23 T^{3} + 24 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_y_ax |
| 17 | $A_4\times C_2$ | \( 1 + 5 T + 43 T^{2} + 129 T^{3} + 43 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.f_br_ez |
| 19 | $A_4\times C_2$ | \( 1 + 10 T + 74 T^{2} + 339 T^{3} + 74 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.k_cw_nb |
| 23 | $A_4\times C_2$ | \( 1 + 20 T^{2} - 91 T^{3} + 20 p T^{4} + p^{3} T^{6} \) | 3.23.a_u_adn |
| 31 | $A_4\times C_2$ | \( 1 + 5 T + 57 T^{2} + 353 T^{3} + 57 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.f_cf_np |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 65 T^{2} + 83 T^{3} + 65 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.d_cn_df |
| 41 | $A_4\times C_2$ | \( 1 + 6 T + 86 T^{2} + 493 T^{3} + 86 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.g_di_sz |
| 43 | $A_4\times C_2$ | \( 1 + 21 T + 255 T^{2} + 2009 T^{3} + 255 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.v_jv_czh |
| 47 | $A_4\times C_2$ | \( 1 + 2 T + 140 T^{2} + 187 T^{3} + 140 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.c_fk_hf |
| 53 | $A_4\times C_2$ | \( 1 - 24 T + 330 T^{2} - 2881 T^{3} + 330 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.ay_ms_aegv |
| 59 | $A_4\times C_2$ | \( 1 + 19 T + 281 T^{2} + 2411 T^{3} + 281 p T^{4} + 19 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.t_kv_dot |
| 61 | $A_4\times C_2$ | \( 1 + 2 T + 84 T^{2} + 257 T^{3} + 84 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.c_dg_jx |
| 67 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{3} \) | 3.67.bh_vs_inh |
| 71 | $A_4\times C_2$ | \( 1 - T + 127 T^{2} - 393 T^{3} + 127 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ab_ex_apd |
| 73 | $A_4\times C_2$ | \( 1 + 8 T + 154 T^{2} + 677 T^{3} + 154 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.i_fy_bab |
| 79 | $A_4\times C_2$ | \( 1 - 14 T + 293 T^{2} - 2268 T^{3} + 293 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ao_lh_adjg |
| 83 | $A_4\times C_2$ | \( 1 + 14 T + 200 T^{2} + 1435 T^{3} + 200 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.o_hs_cdf |
| 89 | $A_4\times C_2$ | \( 1 + 25 T + 466 T^{2} + 4953 T^{3} + 466 p T^{4} + 25 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.z_ry_hin |
| 97 | $A_4\times C_2$ | \( 1 + 4 T + 231 T^{2} + 880 T^{3} + 231 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.e_ix_bhw |
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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
−8.823649672611869542540415138271, −8.495557948303382482000956123900, −8.335465409885209652244035419360, −8.284028997858202041693660204607, −7.50276723456758748610676962853, −7.43148732303967491581870668300, −7.27854335026815919128708244911, −6.58618021968648013526672222445, −6.43020047396734434453098556679, −6.37079677668561607859298765670, −5.98923520766737756175829397726, −5.89905723934871870094660644810, −5.78655308689025560120034542014, −5.15371270392584344915122101105, −5.06880998106689682645097849347, −4.84580047428593082398983467917, −4.29465710314406393465386496822, −3.69019614019029915776147969721, −3.40183790634649067847643934554, −2.73892563133134694981633355845, −2.66060365734896686068647810442, −2.27713585814182180882687826841, −1.85595880764523973402155825677, −1.55007135648515435454103344133, −1.48630710283875076660165321951, 0, 0, 0,
1.48630710283875076660165321951, 1.55007135648515435454103344133, 1.85595880764523973402155825677, 2.27713585814182180882687826841, 2.66060365734896686068647810442, 2.73892563133134694981633355845, 3.40183790634649067847643934554, 3.69019614019029915776147969721, 4.29465710314406393465386496822, 4.84580047428593082398983467917, 5.06880998106689682645097849347, 5.15371270392584344915122101105, 5.78655308689025560120034542014, 5.89905723934871870094660644810, 5.98923520766737756175829397726, 6.37079677668561607859298765670, 6.43020047396734434453098556679, 6.58618021968648013526672222445, 7.27854335026815919128708244911, 7.43148732303967491581870668300, 7.50276723456758748610676962853, 8.284028997858202041693660204607, 8.335465409885209652244035419360, 8.495557948303382482000956123900, 8.823649672611869542540415138271
