| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s + 2·11-s − 2·13-s − 10·17-s − 6·21-s − 2·23-s − 10·27-s + 6·29-s + 3·31-s − 6·33-s − 4·37-s + 6·39-s + 16·41-s − 2·43-s − 12·47-s − 11·49-s + 30·51-s − 6·53-s + 16·59-s + 14·61-s + 12·63-s − 8·67-s + 6·69-s + 10·71-s − 8·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s + 0.603·11-s − 0.554·13-s − 2.42·17-s − 1.30·21-s − 0.417·23-s − 1.92·27-s + 1.11·29-s + 0.538·31-s − 1.04·33-s − 0.657·37-s + 0.960·39-s + 2.49·41-s − 0.304·43-s − 1.75·47-s − 1.57·49-s + 4.20·51-s − 0.824·53-s + 2.08·59-s + 1.79·61-s + 1.51·63-s − 0.977·67-s + 0.722·69-s + 1.18·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 5 | | \( 1 \) | |
| 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.ac_p_abe |
| 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.ac_f_bc |
| 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.c_v_cs |
| 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.ag_bz_ahe |
| 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.e_v_afu |
| 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.aq_gx_acai |
| 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.c_ef_gy |
| 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.aq_jx_aczm |
| 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.i_ex_bso |
| 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.ak_fh_aboc |
| 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.i_fp_bwg |
| 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.i_gx_bug |
| 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.i_kx_cdk |
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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
−7.16509753729126704060703714798, −6.81050061146979210576339666821, −6.60488455285667833541640199265, −6.59695453374672939918661586994, −6.24672863581393780367318530490, −6.06796324364477711571279439619, −5.74957445782221672402415175099, −5.47699086030412870890410538435, −5.33784658543184131863752453512, −5.17591795704975992049807379306, −4.67413833487211197256511095742, −4.56650055600923414777920082755, −4.52571821894659976744303498479, −4.14186024962236756116313895870, −4.12798891057316252027879539318, −3.77941638009035244273997664009, −3.31974278143259443571550461686, −2.87050877325242369540849083642, −2.86742209849130845171282407734, −2.26656936942966263329427527368, −2.14670907105152453514523768264, −1.95770045207357643489317372889, −1.22619070632607291405330109558, −1.21492151372998824060115266064, −1.09079696923655634106944271913, 0, 0, 0,
1.09079696923655634106944271913, 1.21492151372998824060115266064, 1.22619070632607291405330109558, 1.95770045207357643489317372889, 2.14670907105152453514523768264, 2.26656936942966263329427527368, 2.86742209849130845171282407734, 2.87050877325242369540849083642, 3.31974278143259443571550461686, 3.77941638009035244273997664009, 4.12798891057316252027879539318, 4.14186024962236756116313895870, 4.52571821894659976744303498479, 4.56650055600923414777920082755, 4.67413833487211197256511095742, 5.17591795704975992049807379306, 5.33784658543184131863752453512, 5.47699086030412870890410538435, 5.74957445782221672402415175099, 6.06796324364477711571279439619, 6.24672863581393780367318530490, 6.59695453374672939918661586994, 6.60488455285667833541640199265, 6.81050061146979210576339666821, 7.16509753729126704060703714798
