| L(s) = 1 | + 2·4-s + 8·9-s + 16·31-s + 16·36-s − 16·49-s − 8·64-s − 48·71-s + 16·79-s + 30·81-s − 24·89-s + 20·121-s + 32·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 4-s + 8/3·9-s + 2.87·31-s + 8/3·36-s − 2.28·49-s − 64-s − 5.69·71-s + 1.80·79-s + 10/3·81-s − 2.54·89-s + 1.81·121-s + 2.87·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.582154265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.582154265\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) | |
| 5 | | \( 1 \) | |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_ai_a_bi |
| 7 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_q_a_gg |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_au_a_ne |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.13.a_aca_a_bna |
| 17 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_u_a_bac |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) | 4.19.a_aca_a_cbu |
| 23 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_dc_a_dyg |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.29.a_aem_a_hmc |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.31.aq_im_acpc_rso |
| 37 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_ae_a_ebm |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.41.a_gi_a_oxy |
| 43 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_afg_a_mic |
| 47 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_dc_a_ixm |
| 53 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_afs_a_qks |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_au_a_klq |
| 61 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_aim_a_bcxq |
| 67 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_aiy_a_bhew |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) | 4.71.bw_bse_zjc_jwxq |
| 73 | $C_2^2$ | \( ( 1 + 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_jk_a_bluk |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.79.aq_pw_afzs_dafm |
| 83 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_afg_a_bbfu |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.89.y_wa_kts_ezco |
| 97 | $C_2^2$ | \( ( 1 + 170 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_nc_a_cspi |
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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
−9.034863629622467509287143467742, −8.832145599322449668914095871666, −8.695699550072317462418536656108, −8.282645942650435356617524267475, −7.85707682480165890286615358516, −7.73282640100581079560050460003, −7.42487511676325838299668484747, −7.32798481457538144726195690871, −6.82943163525927008037660426903, −6.76710428352093850918858975959, −6.35968494916170159000235810775, −6.27815644379175218601079071807, −6.00381366325380480048907955818, −5.28444214105113213666722943446, −5.24511594469597190508984221240, −4.53433247909762088506813914182, −4.47236384871779669162619044799, −4.31314495397481653900372926294, −3.97974849598179258217987595414, −3.15822956043620498674773716229, −3.07148660936987295083142404534, −2.65329971194560918062265518434, −1.92664439541243910113160453401, −1.59781487526740669129278210192, −1.18281765734069650974819840474,
1.18281765734069650974819840474, 1.59781487526740669129278210192, 1.92664439541243910113160453401, 2.65329971194560918062265518434, 3.07148660936987295083142404534, 3.15822956043620498674773716229, 3.97974849598179258217987595414, 4.31314495397481653900372926294, 4.47236384871779669162619044799, 4.53433247909762088506813914182, 5.24511594469597190508984221240, 5.28444214105113213666722943446, 6.00381366325380480048907955818, 6.27815644379175218601079071807, 6.35968494916170159000235810775, 6.76710428352093850918858975959, 6.82943163525927008037660426903, 7.32798481457538144726195690871, 7.42487511676325838299668484747, 7.73282640100581079560050460003, 7.85707682480165890286615358516, 8.282645942650435356617524267475, 8.695699550072317462418536656108, 8.832145599322449668914095871666, 9.034863629622467509287143467742
