| L(s) = 1 | + 6·3-s − 4-s + 5·5-s − 6·7-s − 3·8-s + 15·9-s − 11-s − 6·12-s − 2·13-s + 30·15-s − 16-s + 9·17-s + 3·19-s − 5·20-s − 36·21-s − 18·24-s + 12·25-s + 8·27-s + 6·28-s + 3·29-s − 31-s + 6·32-s − 6·33-s − 30·35-s − 15·36-s + 3·37-s − 12·39-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 1/2·4-s + 2.23·5-s − 2.26·7-s − 1.06·8-s + 5·9-s − 0.301·11-s − 1.73·12-s − 0.554·13-s + 7.74·15-s − 1/4·16-s + 2.18·17-s + 0.688·19-s − 1.11·20-s − 7.85·21-s − 3.67·24-s + 12/5·25-s + 1.53·27-s + 1.13·28-s + 0.557·29-s − 0.179·31-s + 1.06·32-s − 1.04·33-s − 5.07·35-s − 5/2·36-s + 0.493·37-s − 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3442951 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3442951 ^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.261852036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.261852036\) |
| \(L(\frac{3}{2})\) |
|
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 | 151 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 2 | $S_4\times C_2$ | \( 1 + T^{2} + 3 T^{3} + p T^{4} + p^{3} T^{6} \) | 3.2.a_b_d |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.3.ag_v_abs |
| 5 | $S_4\times C_2$ | \( 1 - p T + 13 T^{2} - p^{2} T^{3} + 13 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) | 3.5.af_n_az |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) | 3.7.g_bh_do |
| 11 | $S_4\times C_2$ | \( 1 + T + 13 T^{2} + 47 T^{3} + 13 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.11.b_n_bv |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 7 T^{2} + 28 T^{3} + 7 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.c_h_bc |
| 17 | $S_4\times C_2$ | \( 1 - 9 T + 73 T^{2} - 321 T^{3} + 73 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.aj_cv_amj |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 21 T^{2} - 33 T^{3} + 21 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.ad_v_abh |
| 23 | $S_4\times C_2$ | \( 1 + 49 T^{2} + 24 T^{3} + 49 p T^{4} + p^{3} T^{6} \) | 3.23.a_bx_y |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 25 T^{2} - 303 T^{3} + 25 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ad_z_alr |
| 31 | $S_4\times C_2$ | \( 1 + T + 85 T^{2} + 59 T^{3} + 85 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.31.b_dh_ch |
| 37 | $S_4\times C_2$ | \( 1 - 3 T + 69 T^{2} - 7 p T^{3} + 69 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ad_cr_ajz |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) | 3.41.a_et_a |
| 43 | $S_4\times C_2$ | \( 1 + T + 121 T^{2} + 83 T^{3} + 121 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.43.b_er_df |
| 47 | $S_4\times C_2$ | \( 1 + 13 T + 193 T^{2} + 1283 T^{3} + 193 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.n_hl_bxj |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 15 T^{2} - 12 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.g_p_am |
| 59 | $S_4\times C_2$ | \( 1 - 23 T + 345 T^{2} - 3101 T^{3} + 345 p T^{4} - 23 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ax_nh_aeph |
| 61 | $S_4\times C_2$ | \( 1 + 8 T + 71 T^{2} + 656 T^{3} + 71 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.i_ct_zg |
| 67 | $S_4\times C_2$ | \( 1 - 2 T + 69 T^{2} - 196 T^{3} + 69 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ac_cr_aho |
| 71 | $S_4\times C_2$ | \( 1 + 193 T^{2} - 24 T^{3} + 193 p T^{4} + p^{3} T^{6} \) | 3.71.a_hl_ay |
| 73 | $S_4\times C_2$ | \( 1 - 10 T + 3 p T^{2} - 1388 T^{3} + 3 p^{2} T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ak_il_acbk |
| 79 | $S_4\times C_2$ | \( 1 + 26 T + 429 T^{2} + 4468 T^{3} + 429 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ba_qn_gpw |
| 83 | $S_4\times C_2$ | \( 1 - 28 T + 421 T^{2} - 4352 T^{3} + 421 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.abc_qf_aglk |
| 89 | $S_4\times C_2$ | \( 1 - 36 T + 679 T^{2} - 7872 T^{3} + 679 p T^{4} - 36 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.abk_bad_alqu |
| 97 | $S_4\times C_2$ | \( 1 + 5 T + 53 T^{2} + 5 T^{3} + 53 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.f_cb_f |
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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
−11.81193230865163819958309663868, −11.52009906169048716366628219826, −10.64109079219684365159506337741, −10.12000632852137901887870903661, −9.771146738962244210807563886805, −9.763216389754272012425876758491, −9.596360478784579741246088181297, −9.357105943568172504003842875613, −8.926364582069228686377422534356, −8.830479524735328428323480504376, −8.119085420415903435014939465545, −8.042365199299374944945461180041, −7.67557453791905362693752425491, −7.02709024980959392033863132334, −6.47185043024457966935105384358, −6.22307463988278107453548334917, −5.82778248384247945664751864632, −5.36270242566125028937965701884, −4.93388420803862514272910086238, −3.61192629299886806621836297533, −3.58564686044922981385608613286, −3.02096844302101234085019610087, −2.91890584097296150406560946324, −2.46494606590211108372264941213, −1.84665339994621814271803417974,
1.84665339994621814271803417974, 2.46494606590211108372264941213, 2.91890584097296150406560946324, 3.02096844302101234085019610087, 3.58564686044922981385608613286, 3.61192629299886806621836297533, 4.93388420803862514272910086238, 5.36270242566125028937965701884, 5.82778248384247945664751864632, 6.22307463988278107453548334917, 6.47185043024457966935105384358, 7.02709024980959392033863132334, 7.67557453791905362693752425491, 8.042365199299374944945461180041, 8.119085420415903435014939465545, 8.830479524735328428323480504376, 8.926364582069228686377422534356, 9.357105943568172504003842875613, 9.596360478784579741246088181297, 9.763216389754272012425876758491, 9.771146738962244210807563886805, 10.12000632852137901887870903661, 10.64109079219684365159506337741, 11.52009906169048716366628219826, 11.81193230865163819958309663868
