| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 4·6-s − 4·7-s − 8-s − 2·9-s + 11-s + 2·12-s + 3·13-s − 8·14-s − 5·16-s − 14·17-s − 4·18-s − 8·21-s + 2·22-s − 8·23-s − 2·24-s + 6·26-s − 9·27-s − 4·28-s + 5·29-s + 31-s − 8·32-s + 2·33-s − 28·34-s − 2·36-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s − 1.51·7-s − 0.353·8-s − 2/3·9-s + 0.301·11-s + 0.577·12-s + 0.832·13-s − 2.13·14-s − 5/4·16-s − 3.39·17-s − 0.942·18-s − 1.74·21-s + 0.426·22-s − 1.66·23-s − 0.408·24-s + 1.17·26-s − 1.73·27-s − 0.755·28-s + 0.928·29-s + 0.179·31-s − 1.41·32-s + 0.348·33-s − 4.80·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 5 | | \( 1 \) | |
| 19 | | \( 1 \) | |
| good | 2 | $A_4\times C_2$ | \( 1 - p T + 3 T^{2} - 3 T^{3} + 3 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.2.ac_d_ad |
| 3 | $A_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - 7 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.ac_g_ah |
| 7 | $A_4\times C_2$ | \( 1 + 4 T + 22 T^{2} + 55 T^{3} + 22 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.e_w_cd |
| 11 | $A_4\times C_2$ | \( 1 - T + 29 T^{2} - 23 T^{3} + 29 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ab_bd_ax |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 3 T^{2} + 25 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ad_d_z |
| 17 | $A_4\times C_2$ | \( 1 + 14 T + 112 T^{2} + 555 T^{3} + 112 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.o_ei_vj |
| 23 | $A_4\times C_2$ | \( 1 + 8 T + 60 T^{2} + 243 T^{3} + 60 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.i_ci_jj |
| 29 | $A_4\times C_2$ | \( 1 - 5 T + 91 T^{2} - 285 T^{3} + 91 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.af_dn_akz |
| 31 | $A_4\times C_2$ | \( 1 - T + 63 T^{2} - 115 T^{3} + 63 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ab_cl_ael |
| 37 | $A_4\times C_2$ | \( 1 - 5 T + p T^{2} + 25 T^{3} + p^{2} T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.af_bl_z |
| 41 | $A_4\times C_2$ | \( 1 + T + p T^{2} - 73 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.41.b_bp_acv |
| 43 | $A_4\times C_2$ | \( 1 - 5 T + 16 T^{2} - 113 T^{3} + 16 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.af_q_aej |
| 47 | $A_4\times C_2$ | \( 1 + 9 T + 77 T^{2} + 535 T^{3} + 77 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.j_cz_up |
| 53 | $A_4\times C_2$ | \( 1 - 31 T + 462 T^{2} - 4191 T^{3} + 462 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.abf_ru_agff |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 137 T^{2} - 508 T^{3} + 137 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ag_fh_ato |
| 61 | $A_4\times C_2$ | \( 1 - 3 T + 173 T^{2} - 367 T^{3} + 173 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ad_gr_aod |
| 67 | $A_4\times C_2$ | \( 1 + 13 T + 3 p T^{2} + 1573 T^{3} + 3 p^{2} T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.n_ht_cin |
| 71 | $A_4\times C_2$ | \( 1 + 7 T + 212 T^{2} + 947 T^{3} + 212 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.h_ie_bkl |
| 73 | $A_4\times C_2$ | \( 1 + T + 189 T^{2} + 199 T^{3} + 189 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.73.b_hh_hr |
| 79 | $A_4\times C_2$ | \( 1 - 18 T + 254 T^{2} - 31 p T^{3} + 254 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.as_ju_adqf |
| 83 | $A_4\times C_2$ | \( 1 + 3 T - 21 T^{2} + 629 T^{3} - 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.d_av_yf |
| 89 | $A_4\times C_2$ | \( 1 - 20 T + 318 T^{2} - 3435 T^{3} + 318 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.au_mg_afcd |
| 97 | $A_4\times C_2$ | \( 1 + 13 T + 3 p T^{2} + 2353 T^{3} + 3 p^{2} T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.n_lf_dmn |
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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
−7.05626848082661800642428140007, −6.80867539496221046428107945587, −6.54093957279716942321221508583, −6.42654056179330457579322123397, −6.31298672222544791922792500939, −6.05550800925812271882508142520, −5.95557867253950828851502747785, −5.45862189760085651979479205082, −5.32101669702446436867846073361, −5.02682788713698226977126429960, −4.68875284717306768217939394593, −4.60978571599731145603590027334, −4.19688507188781376876827279335, −3.95198201199874126923237961211, −3.87400261613288164242751744018, −3.68385552900940692896205546322, −3.57894874540673213385918248255, −2.98064438787443021813651854113, −2.66880575040094026851613959156, −2.65610158870230508281389517066, −2.43541818629676069490079387203, −2.19202879052080751066400050014, −1.90440424774664337175359006892, −1.28805309762109242467223706384, −0.976383068592943837207381505412, 0, 0, 0,
0.976383068592943837207381505412, 1.28805309762109242467223706384, 1.90440424774664337175359006892, 2.19202879052080751066400050014, 2.43541818629676069490079387203, 2.65610158870230508281389517066, 2.66880575040094026851613959156, 2.98064438787443021813651854113, 3.57894874540673213385918248255, 3.68385552900940692896205546322, 3.87400261613288164242751744018, 3.95198201199874126923237961211, 4.19688507188781376876827279335, 4.60978571599731145603590027334, 4.68875284717306768217939394593, 5.02682788713698226977126429960, 5.32101669702446436867846073361, 5.45862189760085651979479205082, 5.95557867253950828851502747785, 6.05550800925812271882508142520, 6.31298672222544791922792500939, 6.42654056179330457579322123397, 6.54093957279716942321221508583, 6.80867539496221046428107945587, 7.05626848082661800642428140007
