| L(s) = 1 | + 3·3-s − 3·4-s + 6·5-s + 8-s + 3·11-s − 9·12-s + 3·13-s + 18·15-s + 3·16-s − 3·17-s + 9·19-s − 18·20-s + 3·24-s + 12·25-s − 10·27-s − 3·29-s + 9·31-s − 6·32-s + 9·33-s + 9·39-s + 6·40-s + 9·41-s − 9·44-s − 3·47-s + 9·48-s − 9·51-s − 9·52-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 3/2·4-s + 2.68·5-s + 0.353·8-s + 0.904·11-s − 2.59·12-s + 0.832·13-s + 4.64·15-s + 3/4·16-s − 0.727·17-s + 2.06·19-s − 4.02·20-s + 0.612·24-s + 12/5·25-s − 1.92·27-s − 0.557·29-s + 1.61·31-s − 1.06·32-s + 1.56·33-s + 1.44·39-s + 0.948·40-s + 1.40·41-s − 1.35·44-s − 0.437·47-s + 1.29·48-s − 1.26·51-s − 1.24·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.160344378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.160344378\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | | \( 1 \) | |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 2 | $A_4\times C_2$ | \( 1 + 3 T^{2} - T^{3} + 3 p T^{4} + p^{3} T^{6} \) | 3.2.a_d_ab |
| 3 | $A_4\times C_2$ | \( 1 - p T + p^{2} T^{2} - 17 T^{3} + p^{3} T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.3.ad_j_ar |
| 5 | $A_4\times C_2$ | \( 1 - 6 T + 24 T^{2} - 61 T^{3} + 24 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.5.ag_y_acj |
| 13 | $A_4\times C_2$ | \( 1 - 3 T + 33 T^{2} - 79 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ad_bh_adb |
| 17 | $A_4\times C_2$ | \( 1 + 3 T + 15 T^{2} - 25 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.d_p_az |
| 19 | $A_4\times C_2$ | \( 1 - 9 T + 75 T^{2} - 351 T^{3} + 75 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.aj_cx_ann |
| 23 | $A_4\times C_2$ | \( 1 + 12 T^{2} - 107 T^{3} + 12 p T^{4} + p^{3} T^{6} \) | 3.23.a_m_aed |
| 29 | $A_4\times C_2$ | \( 1 + 3 T + 51 T^{2} + 225 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.d_bz_ir |
| 31 | $A_4\times C_2$ | \( 1 - 9 T + 117 T^{2} - 577 T^{3} + 117 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.aj_en_awf |
| 37 | $A_4\times C_2$ | \( 1 + 75 T^{2} + 72 T^{3} + 75 p T^{4} + p^{3} T^{6} \) | 3.37.a_cx_cu |
| 41 | $A_4\times C_2$ | \( 1 - 9 T + 129 T^{2} - 739 T^{3} + 129 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.aj_ez_abcl |
| 43 | $A_4\times C_2$ | \( 1 + 120 T^{2} + 9 T^{3} + 120 p T^{4} + p^{3} T^{6} \) | 3.43.a_eq_j |
| 47 | $A_4\times C_2$ | \( 1 + 3 T + 63 T^{2} - 41 T^{3} + 63 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.d_cl_abp |
| 53 | $A_4\times C_2$ | \( 1 - 9 T + 105 T^{2} - 495 T^{3} + 105 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.aj_eb_atb |
| 59 | $A_4\times C_2$ | \( 1 + 84 T^{2} + 19 T^{3} + 84 p T^{4} + p^{3} T^{6} \) | 3.59.a_dg_t |
| 61 | $A_4\times C_2$ | \( 1 - 12 T + 147 T^{2} - 1488 T^{3} + 147 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.am_fr_acfg |
| 67 | $A_4\times C_2$ | \( 1 + 189 T^{2} - 8 T^{3} + 189 p T^{4} + p^{3} T^{6} \) | 3.67.a_hh_ai |
| 71 | $A_4\times C_2$ | \( 1 + 9 T + 123 T^{2} + 477 T^{3} + 123 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.j_et_sj |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 228 T^{2} - 877 T^{3} + 228 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ag_iu_abht |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 123 T^{2} + 511 T^{3} + 123 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.d_et_tr |
| 83 | $A_4\times C_2$ | \( 1 + 15 T + 267 T^{2} + 2223 T^{3} + 267 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.p_kh_dhn |
| 89 | $A_4\times C_2$ | \( 1 + 15 T + 339 T^{2} + 2781 T^{3} + 339 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.p_nb_ecz |
| 97 | $A_4\times C_2$ | \( 1 - 45 T + 963 T^{2} - 12059 T^{3} + 963 p T^{4} - 45 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.abt_blb_arvv |
| 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
−9.590724619340757239077753687657, −9.208642609627153110086849287444, −9.070234680289049851660992534269, −9.030285844044607041270207343199, −8.561105138671921394763248532186, −8.345481287677819273620689181043, −8.127890981537826090359941623706, −7.50545284129951312740717821613, −7.49841074393413470909625021893, −6.75088905224832820031770971635, −6.53755656665418312317099221670, −6.09708071507662225365662524352, −5.93089887958807383868364082568, −5.36440398171834416784743786588, −5.31262599921936768386931761341, −5.11985004261060567397917744596, −4.17174109490220100065343407600, −4.13196420143855307039189995532, −3.82474645514583597149794132363, −2.98306147523787437364138532829, −2.97301603967244538631117214419, −2.47373480484561090507506086664, −1.98196403479871147831307618055, −1.54280918184852114445669431461, −0.923699965710655103292956541919,
0.923699965710655103292956541919, 1.54280918184852114445669431461, 1.98196403479871147831307618055, 2.47373480484561090507506086664, 2.97301603967244538631117214419, 2.98306147523787437364138532829, 3.82474645514583597149794132363, 4.13196420143855307039189995532, 4.17174109490220100065343407600, 5.11985004261060567397917744596, 5.31262599921936768386931761341, 5.36440398171834416784743786588, 5.93089887958807383868364082568, 6.09708071507662225365662524352, 6.53755656665418312317099221670, 6.75088905224832820031770971635, 7.49841074393413470909625021893, 7.50545284129951312740717821613, 8.127890981537826090359941623706, 8.345481287677819273620689181043, 8.561105138671921394763248532186, 9.030285844044607041270207343199, 9.070234680289049851660992534269, 9.208642609627153110086849287444, 9.590724619340757239077753687657
