| L(s) = 1 | − 3-s + 3·4-s + 9-s + 6·11-s − 3·12-s − 3·13-s + 4·16-s + 6·17-s + 16·19-s − 6·23-s + 3·25-s + 4·27-s + 6·29-s − 6·33-s + 3·36-s − 21·37-s + 3·39-s + 3·41-s + 9·43-s + 18·44-s − 12·47-s − 4·48-s − 6·51-s − 9·52-s − 16·57-s − 15·59-s + 9·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 3/2·4-s + 1/3·9-s + 1.80·11-s − 0.866·12-s − 0.832·13-s + 16-s + 1.45·17-s + 3.67·19-s − 1.25·23-s + 3/5·25-s + 0.769·27-s + 1.11·29-s − 1.04·33-s + 1/2·36-s − 3.45·37-s + 0.480·39-s + 0.468·41-s + 1.37·43-s + 2.71·44-s − 1.75·47-s − 0.577·48-s − 0.840·51-s − 1.24·52-s − 2.11·57-s − 1.95·59-s + 1.15·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.856130586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.856130586\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) | |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} ) \) | 4.2.a_ad_a_f |
| 3 | $D_4\times C_2$ | \( 1 + T - 5 T^{3} - 11 T^{4} - 5 p T^{5} + p^{3} T^{7} + p^{4} T^{8} \) | 4.3.b_a_af_al |
| 5 | $D_4\times C_2$ | \( 1 - 3 T^{2} + p T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_ad_a_f |
| 11 | $C_2^2$ | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.ag_f_acc_rc |
| 13 | $D_4\times C_2$ | \( 1 + 3 T - 14 T^{2} - 9 T^{3} + 243 T^{4} - 9 p T^{5} - 14 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.13.d_ao_aj_jj |
| 17 | $D_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 180 T^{3} + 815 T^{4} - 180 p T^{5} + 42 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.17.ag_bq_agy_bfj |
| 23 | $D_4\times C_2$ | \( 1 + 6 T + 2 T^{2} - 72 T^{3} - 201 T^{4} - 72 p T^{5} + 2 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.g_c_acu_aht |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) | 4.29.ag_bt_ahq_wy |
| 31 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_ack_a_egx |
| 37 | $D_4\times C_2$ | \( 1 + 21 T + 242 T^{2} + 1995 T^{3} + 13095 T^{4} + 1995 p T^{5} + 242 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.v_ji_cyt_tjr |
| 41 | $D_4\times C_2$ | \( 1 - 3 T - 28 T^{2} + 135 T^{3} - 681 T^{4} + 135 p T^{5} - 28 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | 4.41.ad_abc_ff_abaf |
| 43 | $D_4\times C_2$ | \( 1 - 9 T - 20 T^{2} - 135 T^{3} + 4869 T^{4} - 135 p T^{5} - 20 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.43.aj_au_aff_hfh |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 147 T^{2} + 1188 T^{3} + 9848 T^{4} + 1188 p T^{5} + 147 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.47.m_fr_bts_oou |
| 53 | $D_4\times C_2$ | \( 1 - 195 T^{2} + 15077 T^{4} - 195 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_ahn_a_whx |
| 59 | $D_4\times C_2$ | \( 1 + 15 T + 56 T^{2} + 765 T^{3} + 11805 T^{4} + 765 p T^{5} + 56 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.p_ce_bdl_rmb |
| 61 | $D_4\times C_2$ | \( 1 - 9 T + 140 T^{2} - 1017 T^{3} + 10695 T^{4} - 1017 p T^{5} + 140 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.aj_fk_abnd_pvj |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) | 4.67.a_ajk_a_bjhu |
| 71 | $D_4\times C_2$ | \( 1 - 6 T + 129 T^{2} - 702 T^{3} + 9500 T^{4} - 702 p T^{5} + 129 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.ag_ez_abba_obk |
| 73 | $D_4\times C_2$ | \( 1 - 6 T + 98 T^{2} - 516 T^{3} + 2943 T^{4} - 516 p T^{5} + 98 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.ag_du_atw_ejf |
| 79 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_aim_a_bkjm |
| 83 | $D_4\times C_2$ | \( 1 - 123 T^{2} + 11129 T^{4} - 123 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_aet_a_qmb |
| 89 | $D_4\times C_2$ | \( 1 - 9 T - 70 T^{2} + 243 T^{3} + 7671 T^{4} + 243 p T^{5} - 70 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) | 4.89.aj_acs_jj_ljb |
| 97 | $C_2^2$ | \( ( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.e_aha_q_bpuh |
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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
−7.05580002773616056346447774631, −6.96527243176668107094630370725, −6.82215898144377648681658369667, −6.62187122817641939276295594383, −6.60958685217028397778544151949, −5.95421823753503279390529249081, −5.75161347675807176244725616771, −5.66518525985555252818569788350, −5.60295273118161610411999865250, −5.11458489919019835727938653118, −4.92917711116148952148543458686, −4.81309086806533586771811716309, −4.47969056211944360115787554080, −4.02891769783034665416661064290, −3.72535949159254494593016221647, −3.55662974029887815902635242914, −3.21181872913884926263204564991, −3.09435498971065470868392046884, −2.89708692441604275153279679853, −2.28239053116736879687334145309, −2.16914259760861771321030769402, −1.50675658336567818424243601900, −1.35357107843611133128419889078, −1.17342542684384856053243610952, −0.60744931870197004409235208046,
0.60744931870197004409235208046, 1.17342542684384856053243610952, 1.35357107843611133128419889078, 1.50675658336567818424243601900, 2.16914259760861771321030769402, 2.28239053116736879687334145309, 2.89708692441604275153279679853, 3.09435498971065470868392046884, 3.21181872913884926263204564991, 3.55662974029887815902635242914, 3.72535949159254494593016221647, 4.02891769783034665416661064290, 4.47969056211944360115787554080, 4.81309086806533586771811716309, 4.92917711116148952148543458686, 5.11458489919019835727938653118, 5.60295273118161610411999865250, 5.66518525985555252818569788350, 5.75161347675807176244725616771, 5.95421823753503279390529249081, 6.60958685217028397778544151949, 6.62187122817641939276295594383, 6.82215898144377648681658369667, 6.96527243176668107094630370725, 7.05580002773616056346447774631
