| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 5^{4} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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.i_ca_jo_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.e_e_fs_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.ai_ie_abum_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.bg_ye_lsa_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
−6.42923693215025745671276594505, −5.79595898101876882548990355133, −5.77816671548749789423729317922, −5.73000325964941908808910208137, −5.58804078287359237675853671123, −5.09670742415856280564611019489, −5.08421302825361014193157930749, −5.07987833760681028194605937766, −5.03957674117244925082029894186, −4.41603669230455896527375165638, −4.40033513957311281029282828341, −4.19577015547146930444025212853, −3.95960790776173936240380933623, −3.41134984426035038667153129393, −3.18140653713423558931918401319, −3.14405675082495069401753025483, −3.10365045870140758492841391206, −2.65184180345953298763855805360, −2.63053148133801498952035335784, −2.28798554570515309578857698590, −2.18848216461085286451741022818, −1.81620609391331703388951221371, −1.56891042208191108395515047001, −1.40837318112986022970647233216, −1.15379187216102418574499136718, 0, 0, 0, 0,
1.15379187216102418574499136718, 1.40837318112986022970647233216, 1.56891042208191108395515047001, 1.81620609391331703388951221371, 2.18848216461085286451741022818, 2.28798554570515309578857698590, 2.63053148133801498952035335784, 2.65184180345953298763855805360, 3.10365045870140758492841391206, 3.14405675082495069401753025483, 3.18140653713423558931918401319, 3.41134984426035038667153129393, 3.95960790776173936240380933623, 4.19577015547146930444025212853, 4.40033513957311281029282828341, 4.41603669230455896527375165638, 5.03957674117244925082029894186, 5.07987833760681028194605937766, 5.08421302825361014193157930749, 5.09670742415856280564611019489, 5.58804078287359237675853671123, 5.73000325964941908808910208137, 5.77816671548749789423729317922, 5.79595898101876882548990355133, 6.42923693215025745671276594505
