| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s − 2·7-s + 2·8-s + 4·9-s − 4·10-s − 2·11-s + 8·13-s − 4·14-s + 8·17-s + 8·18-s + 2·19-s − 6·20-s − 4·22-s + 14·23-s + 11·25-s + 16·26-s − 6·28-s − 32·29-s + 4·31-s − 6·32-s + 16·34-s + 4·35-s + 12·36-s − 4·37-s + 4·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s − 0.755·7-s + 0.707·8-s + 4/3·9-s − 1.26·10-s − 0.603·11-s + 2.21·13-s − 1.06·14-s + 1.94·17-s + 1.88·18-s + 0.458·19-s − 1.34·20-s − 0.852·22-s + 2.91·23-s + 11/5·25-s + 3.13·26-s − 1.13·28-s − 5.94·29-s + 0.718·31-s − 1.06·32-s + 2.74·34-s + 0.676·35-s + 2·36-s − 0.657·37-s + 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.568232726\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.568232726\) |
| \(L(\frac{3}{2})\) |
|
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 | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) | |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) | |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) | 4.2.ac_b_c_ad |
| 3 | $C_2^3$ | \( 1 - 4 T^{2} + 7 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.3.a_ae_a_h |
| 5 | $C_2^2$ | \( ( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.c_ah_c_cy |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 17 T^{2} - 2 T^{3} + 276 T^{4} - 2 p T^{5} - 17 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.c_ar_ac_kq |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.ai_bo_ahs_bio |
| 17 | $C_2^2$ | \( ( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ai_o_aey_brf |
| 23 | $D_4\times C_2$ | \( 1 - 14 T + 103 T^{2} - 658 T^{3} + 3612 T^{4} - 658 p T^{5} + 103 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.ao_dz_azi_fiy |
| 29 | $D_{4}$ | \( ( 1 + 16 T + 120 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.29.bg_tc_hbk_bttq |
| 31 | $D_4\times C_2$ | \( 1 - 4 T - 32 T^{2} + 56 T^{3} + 847 T^{4} + 56 p T^{5} - 32 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.ae_abg_ce_bgp |
| 37 | $D_4\times C_2$ | \( 1 + 4 T - 12 T^{2} - 184 T^{3} - 1177 T^{4} - 184 p T^{5} - 12 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.e_am_ahc_abth |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.y_oa_fjk_boiw |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 37 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.ae_da_ami_iad |
| 47 | $D_4\times C_2$ | \( 1 - 6 T - 65 T^{2} - 42 T^{3} + 6300 T^{4} - 42 p T^{5} - 65 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.ag_acn_abq_jii |
| 53 | $C_2^3$ | \( 1 - 88 T^{2} + 4935 T^{4} - 88 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_adk_a_hhv |
| 59 | $D_4\times C_2$ | \( 1 + 8 T - 62 T^{2} + 64 T^{3} + 8619 T^{4} + 64 p T^{5} - 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.i_ack_cm_mtn |
| 61 | $D_4\times C_2$ | \( 1 + 18 T + 129 T^{2} + 1314 T^{3} + 14540 T^{4} + 1314 p T^{5} + 129 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.s_ez_byo_vng |
| 67 | $D_4\times C_2$ | \( 1 + 8 T - 78 T^{2} + 64 T^{3} + 11387 T^{4} + 64 p T^{5} - 78 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.67.i_ada_cm_qvz |
| 71 | $D_{4}$ | \( ( 1 + 20 T + 224 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.bo_bgq_rlw_greg |
| 73 | $D_4\times C_2$ | \( 1 + 2 T + 57 T^{2} - 398 T^{3} - 2812 T^{4} - 398 p T^{5} + 57 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.c_cf_api_aeee |
| 79 | $C_2^3$ | \( 1 - 156 T^{2} + 18095 T^{4} - 156 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_aga_a_batz |
| 83 | $D_{4}$ | \( ( 1 - 10 T + 173 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.au_re_ahoy_dlfn |
| 89 | $D_4\times C_2$ | \( 1 - 20 T + 124 T^{2} - 1960 T^{3} + 32655 T^{4} - 1960 p T^{5} + 124 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.au_eu_acxk_bwhz |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 222 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.y_wq_lim_fmbu |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.883549328858188374805037288838, −9.304598682106100222203772593334, −9.119779309087692403071749836691, −8.945936579556195760899228822316, −8.931220263708831600285971858378, −8.389333748925541043607820730144, −7.80194237915347831012503672633, −7.56518497943199463793812953821, −7.29923213739200502502863414631, −7.26981639891202818611628110860, −6.99734465527422129574540958658, −6.55588915516868810988146836408, −6.06326102178278294662486898275, −5.86351719539445935547290385186, −5.61032274618736934319138033502, −5.30256118190374305896238469306, −4.82829740307763781934580890757, −4.67084015104798149042497717611, −3.94320343687399321611910970285, −3.74650297550444543911548339010, −3.23326699797043766790306373864, −3.19583649548903511092662437179, −3.11143406574674225220848696824, −1.62349896887612359264536673210, −1.49640255590901697774916193598,
1.49640255590901697774916193598, 1.62349896887612359264536673210, 3.11143406574674225220848696824, 3.19583649548903511092662437179, 3.23326699797043766790306373864, 3.74650297550444543911548339010, 3.94320343687399321611910970285, 4.67084015104798149042497717611, 4.82829740307763781934580890757, 5.30256118190374305896238469306, 5.61032274618736934319138033502, 5.86351719539445935547290385186, 6.06326102178278294662486898275, 6.55588915516868810988146836408, 6.99734465527422129574540958658, 7.26981639891202818611628110860, 7.29923213739200502502863414631, 7.56518497943199463793812953821, 7.80194237915347831012503672633, 8.389333748925541043607820730144, 8.931220263708831600285971858378, 8.945936579556195760899228822316, 9.119779309087692403071749836691, 9.304598682106100222203772593334, 9.883549328858188374805037288838
