| L(s) = 1 | + 3·3-s + 3·5-s − 2·7-s + 6·9-s − 2·11-s − 4·13-s + 9·15-s − 12·17-s − 4·19-s − 6·21-s + 10·23-s + 6·25-s + 10·27-s − 6·29-s − 3·31-s − 6·33-s − 6·35-s − 20·37-s − 12·39-s − 10·41-s + 4·43-s + 18·45-s + 4·47-s − 11·49-s − 36·51-s − 10·53-s − 6·55-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s − 0.755·7-s + 2·9-s − 0.603·11-s − 1.10·13-s + 2.32·15-s − 2.91·17-s − 0.917·19-s − 1.30·21-s + 2.08·23-s + 6/5·25-s + 1.92·27-s − 1.11·29-s − 0.538·31-s − 1.04·33-s − 1.01·35-s − 3.28·37-s − 1.92·39-s − 1.56·41-s + 0.609·43-s + 2.68·45-s + 0.583·47-s − 1.57·49-s − 5.04·51-s − 1.37·53-s − 0.809·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) | |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) | |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T + 15 T^{2} + 20 T^{3} + 15 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.c_p_u |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + 17 T^{2} + 28 T^{3} + 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.c_r_bc |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 37 T^{2} + 100 T^{3} + 37 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.e_bl_dw |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{3} \) | 3.17.m_dv_se |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + T^{2} - 104 T^{3} + p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.19.e_b_aea |
| 23 | $S_4\times C_2$ | \( 1 - 10 T + 73 T^{2} - 372 T^{3} + 73 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.23.ak_cv_aoi |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 33 T^{2} + 132 T^{3} + 33 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.g_bh_fc |
| 37 | $S_4\times C_2$ | \( 1 + 20 T + 237 T^{2} + 1724 T^{3} + 237 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.u_jd_coi |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 127 T^{2} + 732 T^{3} + 127 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.41.k_ex_bce |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 65 T^{2} - 216 T^{3} + 65 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.ae_cn_aii |
| 47 | $S_4\times C_2$ | \( 1 - 4 T + 129 T^{2} - 360 T^{3} + 129 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.ae_ez_anw |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 127 T^{2} + 708 T^{3} + 127 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.53.k_ex_bbg |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 43 T^{2} + 420 T^{3} + 43 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.c_br_qe |
| 61 | $S_4\times C_2$ | \( 1 + 4 T + 27 T^{2} - 200 T^{3} + 27 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.e_bb_ahs |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 147 T^{2} + 588 T^{3} + 147 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.g_fr_wq |
| 71 | $S_4\times C_2$ | \( 1 + 10 T + 111 T^{2} + 732 T^{3} + 111 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.k_eh_bce |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 197 T^{2} + 892 T^{3} + 197 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.g_hp_bii |
| 79 | $S_4\times C_2$ | \( 1 - 8 T + 193 T^{2} - 1280 T^{3} + 193 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.ai_hl_abxg |
| 83 | $S_4\times C_2$ | \( 1 - 8 T + 253 T^{2} - 1296 T^{3} + 253 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ai_jt_abxw |
| 89 | $S_4\times C_2$ | \( 1 + 16 T + 161 T^{2} + 1316 T^{3} + 161 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.q_gf_byq |
| 97 | $S_4\times C_2$ | \( 1 + 6 T + 191 T^{2} + 820 T^{3} + 191 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.97.g_hj_bfo |
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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
−7.30325359730032740169912165606, −7.03721004681474078045603723002, −6.82200023088330471181360508654, −6.81851303930212776070107466731, −6.37984772424550409296950448791, −6.31705039635982577649639241981, −6.18530430715905638339535593053, −5.52971682468000795406409730805, −5.28465206462395063982062785153, −5.21879128155046309154367468507, −4.97638346173686068823293772988, −4.62556040337524117604333903749, −4.59236029098982379246693283073, −4.14729927780943954937694103583, −3.75047660677088881442765611862, −3.72550693448195156609188362296, −3.26269077581370340479974431419, −2.99204036940605748888541398777, −2.86538879517345780971172548568, −2.44590549061611524208324639635, −2.27017485682445314917740441895, −2.24405983883526631766054893315, −1.57993665875642817096039327731, −1.46455008157654512419704250021, −1.41018763255481810499867960162, 0, 0, 0,
1.41018763255481810499867960162, 1.46455008157654512419704250021, 1.57993665875642817096039327731, 2.24405983883526631766054893315, 2.27017485682445314917740441895, 2.44590549061611524208324639635, 2.86538879517345780971172548568, 2.99204036940605748888541398777, 3.26269077581370340479974431419, 3.72550693448195156609188362296, 3.75047660677088881442765611862, 4.14729927780943954937694103583, 4.59236029098982379246693283073, 4.62556040337524117604333903749, 4.97638346173686068823293772988, 5.21879128155046309154367468507, 5.28465206462395063982062785153, 5.52971682468000795406409730805, 6.18530430715905638339535593053, 6.31705039635982577649639241981, 6.37984772424550409296950448791, 6.81851303930212776070107466731, 6.82200023088330471181360508654, 7.03721004681474078045603723002, 7.30325359730032740169912165606
