| L(s) = 1 | − 2·3-s − 3·5-s + 4·7-s − 9-s − 10·11-s + 2·13-s + 6·15-s + 4·17-s − 10·19-s − 8·21-s + 4·23-s + 6·25-s + 4·27-s + 3·29-s − 2·31-s + 20·33-s − 12·35-s − 4·39-s − 10·41-s − 10·43-s + 3·45-s − 2·47-s − 5·49-s − 8·51-s − 6·53-s + 30·55-s + 20·57-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 1.51·7-s − 1/3·9-s − 3.01·11-s + 0.554·13-s + 1.54·15-s + 0.970·17-s − 2.29·19-s − 1.74·21-s + 0.834·23-s + 6/5·25-s + 0.769·27-s + 0.557·29-s − 0.359·31-s + 3.48·33-s − 2.02·35-s − 0.640·39-s − 1.56·41-s − 1.52·43-s + 0.447·45-s − 0.291·47-s − 5/7·49-s − 1.12·51-s − 0.824·53-s + 4.04·55-s + 2.64·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 29 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 8 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.3.c_f_i |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 3 p T^{2} - 52 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ae_v_aca |
| 11 | $S_4\times C_2$ | \( 1 + 10 T + 57 T^{2} + 216 T^{3} + 57 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.k_cf_ii |
| 13 | $D_{6}$ | \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_bb_abs |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 47 T^{2} - 116 T^{3} + 47 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ae_bv_aem |
| 19 | $S_4\times C_2$ | \( 1 + 10 T + 85 T^{2} + 400 T^{3} + 85 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.k_dh_pk |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 41 T^{2} - 68 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ae_bp_acq |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 17 T^{2} + 240 T^{3} + 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.c_r_jg |
| 37 | $S_4\times C_2$ | \( 1 + 83 T^{2} + 52 T^{3} + 83 p T^{4} + p^{3} T^{6} \) | 3.37.a_df_ca |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 71 T^{2} + 284 T^{3} + 71 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.k_ct_ky |
| 43 | $S_4\times C_2$ | \( 1 + 10 T + 157 T^{2} + 880 T^{3} + 157 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.k_gb_bhw |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 65 T^{2} + 304 T^{3} + 65 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.c_cn_ls |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 155 T^{2} + 596 T^{3} + 155 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.g_fz_wy |
| 59 | $S_4\times C_2$ | \( 1 + 12 T + 81 T^{2} + 8 p T^{3} + 81 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.m_dd_se |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 59 T^{2} + 28 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ag_ch_bc |
| 67 | $S_4\times C_2$ | \( 1 + 101 T^{2} + 4 p T^{3} + 101 p T^{4} + p^{3} T^{6} \) | 3.67.a_dx_ki |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 181 T^{2} - 864 T^{3} + 181 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.ai_gz_abhg |
| 73 | $S_4\times C_2$ | \( 1 - 16 T + 259 T^{2} - 2236 T^{3} + 259 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.aq_jz_adia |
| 79 | $S_4\times C_2$ | \( 1 - 26 T + 429 T^{2} - 4544 T^{3} + 429 p T^{4} - 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.aba_qn_agsu |
| 83 | $S_4\times C_2$ | \( 1 + 4 T + 209 T^{2} + 596 T^{3} + 209 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.e_ib_wy |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 167 T^{2} + 1964 T^{3} + 167 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.k_gl_cxo |
| 97 | $S_4\times C_2$ | \( 1 + 16 T + 331 T^{2} + 3004 T^{3} + 331 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.q_mt_elo |
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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
−7.21111818934395395347052495914, −6.90127628113089327643530524151, −6.61880428747181964254268039246, −6.51664922269858420038144883934, −6.18005045678854074261420748094, −5.91514268796510601083282842243, −5.87024867192622019416877151640, −5.38533980346867051213165764089, −5.11223424164812353405798529470, −5.06544261934661338744587293669, −4.82518982525202320879331446827, −4.80367190903098205122796992764, −4.69528033257288780101510993727, −4.01092969053421184650936694630, −3.96224290072888882105992639602, −3.51696327117268914660329982982, −3.25435377751435962826726748601, −3.13524578082719005363064150330, −2.90837571185281878720138121407, −2.33175694874897185065620897165, −2.08880040926930184913519949473, −2.06784751789254767616487440929, −1.56043056495252725939551589693, −1.03581827057950743278616318430, −0.805295987298171723495081094035, 0, 0, 0,
0.805295987298171723495081094035, 1.03581827057950743278616318430, 1.56043056495252725939551589693, 2.06784751789254767616487440929, 2.08880040926930184913519949473, 2.33175694874897185065620897165, 2.90837571185281878720138121407, 3.13524578082719005363064150330, 3.25435377751435962826726748601, 3.51696327117268914660329982982, 3.96224290072888882105992639602, 4.01092969053421184650936694630, 4.69528033257288780101510993727, 4.80367190903098205122796992764, 4.82518982525202320879331446827, 5.06544261934661338744587293669, 5.11223424164812353405798529470, 5.38533980346867051213165764089, 5.87024867192622019416877151640, 5.91514268796510601083282842243, 6.18005045678854074261420748094, 6.51664922269858420038144883934, 6.61880428747181964254268039246, 6.90127628113089327643530524151, 7.21111818934395395347052495914
