| L(s) = 1 | − 3·2-s + 6·4-s + 6·5-s + 6·7-s − 10·8-s − 6·9-s − 18·10-s + 6·11-s + 6·13-s − 18·14-s + 15·16-s + 3·17-s + 18·18-s + 36·20-s − 18·22-s + 9·25-s − 18·26-s + 27-s + 36·28-s − 12·29-s + 6·31-s − 21·32-s − 9·34-s + 36·35-s − 36·36-s + 6·37-s − 60·40-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 2.68·5-s + 2.26·7-s − 3.53·8-s − 2·9-s − 5.69·10-s + 1.80·11-s + 1.66·13-s − 4.81·14-s + 15/4·16-s + 0.727·17-s + 4.24·18-s + 8.04·20-s − 3.83·22-s + 9/5·25-s − 3.53·26-s + 0.192·27-s + 6.80·28-s − 2.22·29-s + 1.07·31-s − 3.71·32-s − 1.54·34-s + 6.08·35-s − 6·36-s + 0.986·37-s − 9.48·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.648945418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.648945418\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) | |
| 19 | | \( 1 \) | |
| good | 3 | $A_4\times C_2$ | \( 1 + 2 p T^{2} - T^{3} + 2 p^{2} T^{4} + p^{3} T^{6} \) | 3.3.a_g_ab |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) | 3.5.ag_bb_acq |
| 7 | $A_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 60 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.7.ag_v_aci |
| 11 | $A_4\times C_2$ | \( 1 - 6 T + 36 T^{2} - 113 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.11.ag_bk_aej |
| 13 | $A_4\times C_2$ | \( 1 - 6 T + 3 p T^{2} - 148 T^{3} + 3 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.13.ag_bn_afs |
| 17 | $A_4\times C_2$ | \( 1 - 3 T + 6 T^{2} + 9 T^{3} + 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) | 3.17.ad_g_j |
| 23 | $A_4\times C_2$ | \( 1 + 57 T^{2} - 8 T^{3} + 57 p T^{4} + p^{3} T^{6} \) | 3.23.a_cf_ai |
| 29 | $A_4\times C_2$ | \( 1 + 12 T + 99 T^{2} + 544 T^{3} + 99 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.29.m_dv_uy |
| 31 | $A_4\times C_2$ | \( 1 - 6 T + 69 T^{2} - 380 T^{3} + 69 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.31.ag_cr_aoq |
| 37 | $A_4\times C_2$ | \( 1 - 6 T + 87 T^{2} - 308 T^{3} + 87 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.37.ag_dj_alw |
| 41 | $A_4\times C_2$ | \( 1 + 120 T^{2} + T^{3} + 120 p T^{4} + p^{3} T^{6} \) | 3.41.a_eq_b |
| 43 | $A_4\times C_2$ | \( 1 - 9 T + 144 T^{2} - 757 T^{3} + 144 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.43.aj_fo_abdd |
| 47 | $A_4\times C_2$ | \( 1 + 6 T + 69 T^{2} + 268 T^{3} + 69 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.47.g_cr_ki |
| 53 | $A_4\times C_2$ | \( 1 + 75 T^{2} - 136 T^{3} + 75 p T^{4} + p^{3} T^{6} \) | 3.53.a_cx_afg |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 168 T^{2} - 711 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.59.ag_gm_abbj |
| 61 | $A_4\times C_2$ | \( 1 - 18 T + 279 T^{2} - 2348 T^{3} + 279 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.61.as_kt_admi |
| 67 | $A_4\times C_2$ | \( 1 - 18 T + 282 T^{2} - 2493 T^{3} + 282 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.67.as_kw_adrx |
| 71 | $A_4\times C_2$ | \( 1 - 12 T + 225 T^{2} - 1552 T^{3} + 225 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.71.am_ir_achs |
| 73 | $A_4\times C_2$ | \( 1 - 12 T + 156 T^{2} - 1695 T^{3} + 156 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) | 3.73.am_ga_acnf |
| 79 | $A_4\times C_2$ | \( 1 - 18 T + 333 T^{2} - 2980 T^{3} + 333 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) | 3.79.as_mv_aekq |
| 83 | $A_4\times C_2$ | \( 1 - 6 T + 222 T^{2} - 945 T^{3} + 222 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) | 3.83.ag_io_abkj |
| 89 | $A_4\times C_2$ | \( 1 + 9 T + 150 T^{2} + 621 T^{3} + 150 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) | 3.89.j_fu_xx |
| 97 | $A_4\times C_2$ | \( 1 + 288 T^{2} + T^{3} + 288 p T^{4} + p^{3} T^{6} \) | 3.97.a_lc_b |
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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
−9.418073384910046762245221473982, −8.893820208237079666017834701928, −8.816405147039691545437046672649, −8.585350169219609860938275419635, −8.028507030149892909083744894899, −8.010489372792661589567420402807, −7.922488669839931201849838061270, −7.40992249267164460523629010721, −6.79494444937683517605758291550, −6.55166341077350550769152703559, −6.22164820933960032027253201021, −6.07340668907106700730282520164, −5.90566275473793991767685907823, −5.35167074208499164514815347095, −5.22886524434935809283915471190, −5.11040012581099276903792840414, −4.02210022282092120725833698475, −3.65103363340462378477404886649, −3.63216734903182382350584633602, −2.54785592444407682402047754685, −2.23193275583489524664608929567, −2.20934183984285395708508559718, −1.65008248790644671020184690798, −1.10910445578557446570450185485, −1.05279665612880239934842221626,
1.05279665612880239934842221626, 1.10910445578557446570450185485, 1.65008248790644671020184690798, 2.20934183984285395708508559718, 2.23193275583489524664608929567, 2.54785592444407682402047754685, 3.63216734903182382350584633602, 3.65103363340462378477404886649, 4.02210022282092120725833698475, 5.11040012581099276903792840414, 5.22886524434935809283915471190, 5.35167074208499164514815347095, 5.90566275473793991767685907823, 6.07340668907106700730282520164, 6.22164820933960032027253201021, 6.55166341077350550769152703559, 6.79494444937683517605758291550, 7.40992249267164460523629010721, 7.922488669839931201849838061270, 8.010489372792661589567420402807, 8.028507030149892909083744894899, 8.585350169219609860938275419635, 8.816405147039691545437046672649, 8.893820208237079666017834701928, 9.418073384910046762245221473982
