| L(s) = 1 | + 3·5-s − 2·7-s − 2·11-s + 2·13-s − 10·17-s − 2·23-s + 6·25-s − 6·29-s + 3·31-s − 6·35-s + 4·37-s − 16·41-s + 2·43-s − 12·47-s − 11·49-s − 6·53-s − 6·55-s − 16·59-s + 14·61-s + 6·65-s + 8·67-s − 10·71-s + 8·73-s + 4·77-s − 16·79-s − 26·83-s − 30·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.755·7-s − 0.603·11-s + 0.554·13-s − 2.42·17-s − 0.417·23-s + 6/5·25-s − 1.11·29-s + 0.538·31-s − 1.01·35-s + 0.657·37-s − 2.49·41-s + 0.304·43-s − 1.75·47-s − 1.57·49-s − 0.824·53-s − 0.809·55-s − 2.08·59-s + 1.79·61-s + 0.744·65-s + 0.977·67-s − 1.18·71-s + 0.936·73-s + 0.455·77-s − 1.80·79-s − 2.85·83-s − 3.25·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 31 | $C_1$ | \( ( 1 - T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T + 15 T^{2} + 30 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.c_p_be |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} - 28 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.c_f_abc |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 21 T^{2} - 70 T^{3} + 21 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ac_v_acs |
| 17 | $S_4\times C_2$ | \( 1 + 10 T + 67 T^{2} + 304 T^{3} + 67 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.k_cp_ls |
| 19 | $S_4\times C_2$ | \( 1 + 25 T^{2} + 32 T^{3} + 25 p T^{4} + p^{3} T^{6} \) | 3.19.a_z_bg |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 29 T^{2} + 8 T^{3} + 29 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.c_bd_i |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 51 T^{2} + 186 T^{3} + 51 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.g_bz_he |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 21 T^{2} + 150 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ae_v_fu |
| 41 | $S_4\times C_2$ | \( 1 + 16 T + 179 T^{2} + 1360 T^{3} + 179 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.q_gx_cai |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 109 T^{2} - 180 T^{3} + 109 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ac_ef_agy |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 181 T^{2} + 1164 T^{3} + 181 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.m_gz_bsu |
| 53 | $S_4\times C_2$ | \( 1 + 6 T + 163 T^{2} + 624 T^{3} + 163 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.g_gh_ya |
| 59 | $S_4\times C_2$ | \( 1 + 16 T + 257 T^{2} + 2014 T^{3} + 257 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.q_jx_czm |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 163 T^{2} - 1236 T^{3} + 163 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.ao_gh_abvo |
| 67 | $S_4\times C_2$ | \( 1 - 8 T + 127 T^{2} - 1158 T^{3} + 127 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.ai_ex_abso |
| 71 | $S_4\times C_2$ | \( 1 + 10 T + 137 T^{2} + 1042 T^{3} + 137 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.k_fh_boc |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 145 T^{2} - 1254 T^{3} + 145 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ai_fp_abwg |
| 79 | $S_4\times C_2$ | \( 1 + 16 T + 265 T^{2} + 2532 T^{3} + 265 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.q_kf_dtk |
| 83 | $S_4\times C_2$ | \( 1 + 26 T + 457 T^{2} + 4832 T^{3} + 457 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ba_rp_hdw |
| 89 | $S_4\times C_2$ | \( 1 - 8 T + 179 T^{2} - 1202 T^{3} + 179 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ai_gx_abug |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 283 T^{2} - 1440 T^{3} + 283 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.ai_kx_acdk |
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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
−7.66564403555719748688104664408, −7.19435041086742500474068065374, −6.78138695493028395292062269780, −6.77059950157114479591021908348, −6.56962123895342825510724763724, −6.52133961782170156393349286297, −6.24567446818787926606379634770, −5.84833207203379165875713647275, −5.60138023600608183173706358843, −5.54072604180674349851697439806, −5.01758482632253981528245502032, −4.90362837227305890128052929148, −4.82722019571293167610408595536, −4.28597140123710732057519360331, −4.04325881530403307659156220082, −3.98209696859903786975943892607, −3.30704339038733697450772622244, −3.28592761374417731174094997246, −3.00926511135730184586770457427, −2.43637037427087523429953787128, −2.39255774383223232703610355316, −2.24228256431077491831178261257, −1.48727040084030862466813262509, −1.42759902211248693539539347912, −1.35367298879709028623381766030, 0, 0, 0,
1.35367298879709028623381766030, 1.42759902211248693539539347912, 1.48727040084030862466813262509, 2.24228256431077491831178261257, 2.39255774383223232703610355316, 2.43637037427087523429953787128, 3.00926511135730184586770457427, 3.28592761374417731174094997246, 3.30704339038733697450772622244, 3.98209696859903786975943892607, 4.04325881530403307659156220082, 4.28597140123710732057519360331, 4.82722019571293167610408595536, 4.90362837227305890128052929148, 5.01758482632253981528245502032, 5.54072604180674349851697439806, 5.60138023600608183173706358843, 5.84833207203379165875713647275, 6.24567446818787926606379634770, 6.52133961782170156393349286297, 6.56962123895342825510724763724, 6.77059950157114479591021908348, 6.78138695493028395292062269780, 7.19435041086742500474068065374, 7.66564403555719748688104664408
