| L(s) = 1 | + 3·2-s − 3-s + 6·4-s − 3·6-s + 5·7-s + 10·8-s − 6·9-s − 4·11-s − 6·12-s + 15·14-s + 15·16-s − 10·17-s − 18·18-s − 8·19-s − 5·21-s − 12·22-s + 3·23-s − 10·24-s + 8·27-s + 30·28-s + 3·29-s − 8·31-s + 21·32-s + 4·33-s − 30·34-s − 36·36-s + 8·37-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.577·3-s + 3·4-s − 1.22·6-s + 1.88·7-s + 3.53·8-s − 2·9-s − 1.20·11-s − 1.73·12-s + 4.00·14-s + 15/4·16-s − 2.42·17-s − 4.24·18-s − 1.83·19-s − 1.09·21-s − 2.55·22-s + 0.625·23-s − 2.04·24-s + 1.53·27-s + 5.66·28-s + 0.557·29-s − 1.43·31-s + 3.71·32-s + 0.696·33-s − 5.14·34-s − 6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 5 | | \( 1 \) | |
| 13 | | \( 1 \) | |
| good | 3 | $A_4\times C_2$ | \( 1 + T + 7 T^{2} + 5 T^{3} + 7 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) | 3.3.b_h_f |
| 7 | $A_4\times C_2$ | \( 1 - 5 T + 27 T^{2} - 71 T^{3} + 27 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.af_bb_act |
| 11 | $A_4\times C_2$ | \( 1 + 4 T + 29 T^{2} + 80 T^{3} + 29 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.e_bd_dc |
| 17 | $A_4\times C_2$ | \( 1 + 10 T + 75 T^{2} + 348 T^{3} + 75 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.k_cx_nk |
| 19 | $A_4\times C_2$ | \( 1 + 8 T + 69 T^{2} + 296 T^{3} + 69 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.i_cr_lk |
| 23 | $A_4\times C_2$ | \( 1 - 3 T + p T^{2} + T^{3} + p^{2} T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ad_x_b |
| 29 | $A_4\times C_2$ | \( 1 - 3 T + 83 T^{2} - 175 T^{3} + 83 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.ad_df_agt |
| 31 | $A_4\times C_2$ | \( 1 + 8 T + 49 T^{2} + 152 T^{3} + 49 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.i_bx_fw |
| 37 | $A_4\times C_2$ | \( 1 - 8 T + 95 T^{2} - 528 T^{3} + 95 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ai_dr_aui |
| 41 | $A_4\times C_2$ | \( 1 + 11 T + 49 T^{2} + 75 T^{3} + 49 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.l_bx_cx |
| 43 | $A_4\times C_2$ | \( 1 - T + 99 T^{2} - p T^{3} + 99 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ab_dv_abr |
| 47 | $A_4\times C_2$ | \( 1 - 15 T + 167 T^{2} - 1339 T^{3} + 167 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ap_gl_abzn |
| 53 | $A_4\times C_2$ | \( 1 + 4 T + 43 T^{2} - 144 T^{3} + 43 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.e_br_afo |
| 59 | $A_4\times C_2$ | \( 1 + 24 T + 341 T^{2} + 3176 T^{3} + 341 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.y_nd_ese |
| 61 | $A_4\times C_2$ | \( 1 + 15 T + 167 T^{2} + 1297 T^{3} + 167 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.p_gl_bxx |
| 67 | $A_4\times C_2$ | \( 1 - 5 T + 123 T^{2} - 839 T^{3} + 123 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.af_et_abgh |
| 71 | $A_4\times C_2$ | \( 1 + 32 T + 545 T^{2} + 5656 T^{3} + 545 p T^{4} + 32 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.bg_uz_ijo |
| 73 | $A_4\times C_2$ | \( 1 + 26 T + 379 T^{2} + 3692 T^{3} + 379 p T^{4} + 26 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.ba_op_fma |
| 79 | $A_4\times C_2$ | \( 1 - 2 T + 117 T^{2} + 28 T^{3} + 117 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ac_en_bc |
| 83 | $A_4\times C_2$ | \( 1 - 7 T + 95 T^{2} - 875 T^{3} + 95 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ah_dr_abhr |
| 89 | $A_4\times C_2$ | \( 1 - 17 T + 137 T^{2} - 829 T^{3} + 137 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.ar_fh_abfx |
| 97 | $A_4\times C_2$ | \( 1 + 6 T + 107 T^{2} + 52 T^{3} + 107 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.g_ed_ca |
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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
−7.28630509812147364158799227652, −6.69505900691942699168668227931, −6.65862595277436178851223900483, −6.39496181892641305956165554386, −6.14585949829227486578238240132, −6.01876026787627325961492294124, −5.79693475069551633451222043783, −5.50442891918078297765751406545, −5.37486582519885853355668368138, −5.06204179330375581029066991425, −4.82293791656479309940895820553, −4.58876668897420508315600073724, −4.50831794176930278832691172812, −4.36572628037845051758791881501, −4.15978503190235329077529206710, −3.69022731117469917935044149194, −3.16629734830210133130574252827, −3.13361286710680485066308492444, −2.92114874742559667892189024388, −2.45745314159954335525828708976, −2.39440964351478719755700737479, −2.18323679323464997611626366139, −1.64317721612752377399884655487, −1.54449591292778101667943071351, −1.20140261000468571154143738766, 0, 0, 0,
1.20140261000468571154143738766, 1.54449591292778101667943071351, 1.64317721612752377399884655487, 2.18323679323464997611626366139, 2.39440964351478719755700737479, 2.45745314159954335525828708976, 2.92114874742559667892189024388, 3.13361286710680485066308492444, 3.16629734830210133130574252827, 3.69022731117469917935044149194, 4.15978503190235329077529206710, 4.36572628037845051758791881501, 4.50831794176930278832691172812, 4.58876668897420508315600073724, 4.82293791656479309940895820553, 5.06204179330375581029066991425, 5.37486582519885853355668368138, 5.50442891918078297765751406545, 5.79693475069551633451222043783, 6.01876026787627325961492294124, 6.14585949829227486578238240132, 6.39496181892641305956165554386, 6.65862595277436178851223900483, 6.69505900691942699168668227931, 7.28630509812147364158799227652
