| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.14223788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.14223788\) |
| \(L(\frac{3}{2})\) |
|
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.ac_f_ai |
| 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.ak_cf_aii |
| 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.c_bb_bs |
| 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.ak_dh_apk |
| 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_aca |
| 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.ak_gb_abhw |
| 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.ag_fz_awy |
| 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.am_dd_ase |
| 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.g_ch_abc |
| 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_aki |
| 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.ae_ib_awy |
| 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
−8.185935419091983803497888997942, −7.64550900838644964991957735691, −7.57352455983281808408226487913, −7.29591875022095861133876568528, −6.76187771708589136853220380264, −6.72445440295037861296373105634, −6.66040784633280151740584575729, −6.09188227127690924574333669058, −5.81140395685851545170364816872, −5.51571320238036937783075273215, −5.30025853644153223241245211964, −5.13365505954747778599685966423, −4.82022133134598833559201925454, −4.45360171894571214656115523475, −4.08609254026839541929044220083, −3.73667721452810618475048782909, −3.35211118480778182054904795429, −3.23567460833641794162329489770, −3.08132399823929418053412997854, −2.28488821735697753980577615878, −2.19452430266067776964663909734, −1.87376336417282290375137325130, −1.36879956828588715663350878648, −1.03064419559632938529725774090, −0.957358536807291547678858236720,
0.957358536807291547678858236720, 1.03064419559632938529725774090, 1.36879956828588715663350878648, 1.87376336417282290375137325130, 2.19452430266067776964663909734, 2.28488821735697753980577615878, 3.08132399823929418053412997854, 3.23567460833641794162329489770, 3.35211118480778182054904795429, 3.73667721452810618475048782909, 4.08609254026839541929044220083, 4.45360171894571214656115523475, 4.82022133134598833559201925454, 5.13365505954747778599685966423, 5.30025853644153223241245211964, 5.51571320238036937783075273215, 5.81140395685851545170364816872, 6.09188227127690924574333669058, 6.66040784633280151740584575729, 6.72445440295037861296373105634, 6.76187771708589136853220380264, 7.29591875022095861133876568528, 7.57352455983281808408226487913, 7.64550900838644964991957735691, 8.185935419091983803497888997942
