| L(s) = 1 | − 4·5-s − 8·9-s + 8·13-s − 4·17-s + 10·25-s + 4·29-s + 4·37-s − 24·41-s + 32·45-s − 16·49-s − 8·53-s + 32·61-s − 32·65-s − 20·73-s + 30·81-s + 16·85-s − 24·89-s + 12·97-s + 16·101-s + 16·109-s − 36·113-s − 64·117-s − 40·121-s − 20·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 8/3·9-s + 2.21·13-s − 0.970·17-s + 2·25-s + 0.742·29-s + 0.657·37-s − 3.74·41-s + 4.77·45-s − 2.28·49-s − 1.09·53-s + 4.09·61-s − 3.96·65-s − 2.34·73-s + 10/3·81-s + 1.73·85-s − 2.54·89-s + 1.21·97-s + 1.59·101-s + 1.53·109-s − 3.38·113-s − 5.91·117-s − 3.63·121-s − 1.78·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 3 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_i_a_bi |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_q_a_gg |
| 11 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_bo_a_ys |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.ai_ca_ajo_bpm |
| 17 | $D_{4}$ | \( ( 1 + 2 T + 32 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.e_cq_ho_cow |
| 19 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_cm_a_cpe |
| 23 | $D_4\times C_2$ | \( 1 + 40 T^{2} + 1266 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_bo_a_bws |
| 31 | $D_4\times C_2$ | \( 1 + 96 T^{2} + 4034 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_ds_a_fze |
| 37 | $D_{4}$ | \( ( 1 - 2 T - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ae_e_afs_ems |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.y_ns_ffs_bnbq |
| 43 | $D_4\times C_2$ | \( 1 + 120 T^{2} + 7106 T^{4} + 120 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_eq_a_kni |
| 47 | $D_4\times C_2$ | \( 1 + 104 T^{2} + 5394 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_ea_a_hzm |
| 53 | $D_{4}$ | \( ( 1 + 4 T + 98 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.i_ie_bum_zas |
| 59 | $D_4\times C_2$ | \( 1 + 124 T^{2} + 10374 T^{4} + 124 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_eu_a_pja |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.61.abg_ye_alsa_eekc |
| 67 | $D_4\times C_2$ | \( 1 + 216 T^{2} + 20450 T^{4} + 216 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ii_a_bego |
| 71 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 6870 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_aca_a_keg |
| 73 | $D_{4}$ | \( ( 1 + 10 T + 96 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.u_lg_faa_byzy |
| 79 | $D_4\times C_2$ | \( 1 + 232 T^{2} + 24210 T^{4} + 232 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_iy_a_bjve |
| 83 | $D_4\times C_2$ | \( 1 + 176 T^{2} + 19794 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_gu_a_bdhi |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.y_si_jbk_dyda |
| 97 | $D_{4}$ | \( ( 1 - 6 T + 56 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.am_fs_acsq_bqva |
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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
−5.93093647947755651938719403060, −5.36017565798829155380961291904, −5.32656201380458629577593910272, −5.24647339617212669900711867052, −5.24583167223938616925838278473, −4.78897274335570746134771412535, −4.65371190727534102268401712187, −4.46223009510708653918540893361, −4.42843214636890429079319225752, −3.88834283830987173694091399063, −3.88664652984472413687944297870, −3.75973663644137420288065499369, −3.44484587268188516588285951431, −3.33438960038892067395490283575, −3.28885955271148051883909967273, −3.07078346101298524305230249379, −2.84931862309834702795037475099, −2.40694116386931360786454799051, −2.40261640130947093845261192543, −2.28893454356229757368927316934, −1.90742879028690782037271789220, −1.36803793513989120177744349535, −1.24339687555777964447593986086, −1.19690750603525237738991842744, −0.882406497093187069433029329965, 0, 0, 0, 0,
0.882406497093187069433029329965, 1.19690750603525237738991842744, 1.24339687555777964447593986086, 1.36803793513989120177744349535, 1.90742879028690782037271789220, 2.28893454356229757368927316934, 2.40261640130947093845261192543, 2.40694116386931360786454799051, 2.84931862309834702795037475099, 3.07078346101298524305230249379, 3.28885955271148051883909967273, 3.33438960038892067395490283575, 3.44484587268188516588285951431, 3.75973663644137420288065499369, 3.88664652984472413687944297870, 3.88834283830987173694091399063, 4.42843214636890429079319225752, 4.46223009510708653918540893361, 4.65371190727534102268401712187, 4.78897274335570746134771412535, 5.24583167223938616925838278473, 5.24647339617212669900711867052, 5.32656201380458629577593910272, 5.36017565798829155380961291904, 5.93093647947755651938719403060
