| L(s) = 1 | − 12·9-s + 20·25-s − 28·49-s + 90·81-s + 8·89-s − 24·97-s + 40·113-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4·9-s + 4·25-s − 4·49-s + 10·81-s + 0.847·89-s − 2.43·97-s + 3.76·113-s − 2·121-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 + 2.76·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7687101848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7687101848\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.3.a_m_a_cc |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.5.a_au_a_fu |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.7.a_bc_a_li |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_abk_a_zm |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.17.a_acq_a_cos |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_m_a_bde |
| 23 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_adg_a_eeo |
| 29 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_bc_a_cug |
| 31 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_aca_a_dvy |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.37.a_afs_a_mdy |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.41.a_agi_a_oxy |
| 43 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_adg_a_icc |
| 47 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_m_a_gpi |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.53.a_aie_a_yyg |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.59.a_jc_a_bexi |
| 61 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ga_a_uag |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.67.a_ki_a_bnvy |
| 71 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_ee_a_tfy |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.73.a_alg_a_bvhu |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.79.a_me_a_cdkg |
| 83 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_ajk_a_bqkk |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.89.ai_oq_adfk_cyqw |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.97.y_xg_lpw_frkw |
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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
−7.49206310328733458596295233554, −7.43085729787990434743036634566, −6.89586182135943676285893007269, −6.66529370172828867277198256449, −6.56424233642022652305972842560, −6.29384102618650091583978966784, −6.14691316945869649446209765395, −5.99279671209097779452233564463, −5.45576234071576591774199769293, −5.32450945148550639442760519690, −5.18069740553163020116660004480, −4.94289345502505940406375875477, −4.82317950480503053060113714130, −4.44633595518846311958222797316, −3.98713989765181349218598890260, −3.60747181639442139948900705329, −3.27399509000808236787401489906, −3.07854631299642481174986007264, −2.94783641384089379757192054831, −2.73952863347801191680402888565, −2.42454440144187324820288750054, −1.96881435697811523722056562804, −1.42755677764870465508226377061, −0.865917061485845127582029526068, −0.29144439211631706463719849052,
0.29144439211631706463719849052, 0.865917061485845127582029526068, 1.42755677764870465508226377061, 1.96881435697811523722056562804, 2.42454440144187324820288750054, 2.73952863347801191680402888565, 2.94783641384089379757192054831, 3.07854631299642481174986007264, 3.27399509000808236787401489906, 3.60747181639442139948900705329, 3.98713989765181349218598890260, 4.44633595518846311958222797316, 4.82317950480503053060113714130, 4.94289345502505940406375875477, 5.18069740553163020116660004480, 5.32450945148550639442760519690, 5.45576234071576591774199769293, 5.99279671209097779452233564463, 6.14691316945869649446209765395, 6.29384102618650091583978966784, 6.56424233642022652305972842560, 6.66529370172828867277198256449, 6.89586182135943676285893007269, 7.43085729787990434743036634566, 7.49206310328733458596295233554
