| L(s) = 1 | − 12·13-s − 10·17-s + 12·23-s − 6·25-s − 36·29-s − 32·43-s − 11·49-s − 2·53-s − 10·61-s + 20·101-s − 8·103-s − 16·107-s + 68·113-s − 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 82·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 3.32·13-s − 2.42·17-s + 2.50·23-s − 6/5·25-s − 6.68·29-s − 4.87·43-s − 1.57·49-s − 0.274·53-s − 1.28·61-s + 1.99·101-s − 0.788·103-s − 1.54·107-s + 6.39·113-s − 1.90·121-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 + 6.30·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.008371116467\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.008371116467\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 3 | | \( 1 \) | |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) | |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) | 4.5.a_g_a_l |
| 11 | $C_2^3$ | \( 1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_v_a_mi |
| 17 | $C_2^2$ | \( ( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.k_bp_jq_cfk |
| 19 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \) | 4.19.a_al_a_ajg |
| 23 | $C_2^2$ | \( ( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.am_ck_aqq_egx |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{4} \) | 4.29.bk_xe_iyq_cgwp |
| 31 | $C_2^3$ | \( 1 - 2 T^{2} - 957 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_ac_a_abkv |
| 37 | $C_2^3$ | \( 1 + 38 T^{2} + 75 T^{4} + 38 p^{2} T^{6} + p^{4} T^{8} \) | 4.37.a_bm_a_cx |
| 41 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_aga_a_nzi |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.43.bg_vk_jdo_ctik |
| 47 | $C_2^3$ | \( 1 + 93 T^{2} + 6440 T^{4} + 93 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_dp_a_jns |
| 53 | $C_2^2$ | \( ( 1 + T - 52 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.c_adz_c_mme |
| 59 | $C_2^3$ | \( 1 + 37 T^{2} - 2112 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_bl_a_addg |
| 61 | $C_2^2$ | \( ( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.k_abv_jq_rlk |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_ev_a_qmi |
| 71 | $C_2^2$ | \( ( 1 + 83 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_gk_a_zct |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} ) \) | 4.73.a_bu_a_aetp |
| 79 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_agc_a_bbsd |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.83.a_amu_a_cjdu |
| 89 | $C_2^3$ | \( 1 + 142 T^{2} + 12243 T^{4} + 142 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_fm_a_scx |
| 97 | $C_2^2$ | \( ( 1 - 190 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_aoq_a_ddgg |
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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
−6.09644681802027163038334353435, −5.86300794450663624872007749962, −5.80659737211474216662143625298, −5.36803693751292621215554789689, −5.20216715426150324659138271279, −5.16971492036559544102475897940, −4.82363019653187247934354597680, −4.74560236352687636506665315183, −4.62444942594934255249692525412, −4.41333185881792064401762936105, −4.16091810141942787764780964514, −3.67627942012269483101388883185, −3.46836430647381808699518191536, −3.42705935353958626052492698376, −3.32002540076028728527778498077, −3.03715231457199159560458957743, −2.47304016172795231353342941085, −2.40635628108301663962136646733, −2.05509945244799644978857400313, −1.94744519974559683596632006718, −1.77366890891672691534895392236, −1.66193130306926541428804350803, −1.01961950131187694419633433106, −0.15428230279692695176971500529, −0.05503649991898183196097299724,
0.05503649991898183196097299724, 0.15428230279692695176971500529, 1.01961950131187694419633433106, 1.66193130306926541428804350803, 1.77366890891672691534895392236, 1.94744519974559683596632006718, 2.05509945244799644978857400313, 2.40635628108301663962136646733, 2.47304016172795231353342941085, 3.03715231457199159560458957743, 3.32002540076028728527778498077, 3.42705935353958626052492698376, 3.46836430647381808699518191536, 3.67627942012269483101388883185, 4.16091810141942787764780964514, 4.41333185881792064401762936105, 4.62444942594934255249692525412, 4.74560236352687636506665315183, 4.82363019653187247934354597680, 5.16971492036559544102475897940, 5.20216715426150324659138271279, 5.36803693751292621215554789689, 5.80659737211474216662143625298, 5.86300794450663624872007749962, 6.09644681802027163038334353435
