| L(s) = 1 | + 2·4-s − 5·16-s − 16·19-s + 10·25-s − 28·49-s − 20·64-s − 32·76-s + 20·100-s + 44·121-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 − 56·196-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 4-s − 5/4·16-s − 3.67·19-s + 2·25-s − 4·49-s − 5/2·64-s − 3.67·76-s + 2·100-s + 4·121-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 + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s − 4·196-s + 0.0712·197-s + 0.0708·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7296089450\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7296089450\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) | |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) | 4.2.a_ac_a_j |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.7.a_bc_a_li |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.11.a_abs_a_bby |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.13.a_aca_a_bna |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.19.q_gq_bsy_izm |
| 23 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_acq_a_dhe |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.29.a_aem_a_hmc |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) | 4.31.a_ae_a_cwc |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.37.a_fs_a_mdy |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.41.a_agi_a_oxy |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.43.a_agq_a_qks |
| 47 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_bc_a_gvm |
| 53 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_gq_a_tgo |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.59.a_jc_a_bexi |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) | 4.61.a_jc_a_bfpu |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.67.a_aki_a_bnvy |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.71.a_aky_a_bsti |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.73.a_lg_a_bvhu |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) | 4.79.a_aho_a_bgrm |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_alw_a_cdmc |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.89.a_ns_a_cshy |
| 97 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.97.a_oy_a_dfni |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.34505160909897297023176369342, −7.20156858310878440825750243119, −6.85157231697169902243924584688, −6.50881405915631865574270683295, −6.50388505659398930492044929706, −6.41799675049240358535866052566, −6.35326814101051264697107161697, −5.91253508235323922810484612772, −5.44596548826748773317403988706, −5.43649535240881735650916149932, −4.78610251837001875769641906784, −4.75672360823448726700323532419, −4.62037289882485831277851789444, −4.37402390193113619447820536037, −3.98017616975734977729022163462, −3.87660901936360952201738372395, −3.07640697218265104315149821334, −3.07573977469608401457911338449, −3.06549157842465894591298728496, −2.29079055062594220362084331286, −2.22716966263648378073160227669, −1.89121523431789211349572207683, −1.71246089048032811731635317542, −1.04703311886733021261851660302, −0.21298324772350881072964902859,
0.21298324772350881072964902859, 1.04703311886733021261851660302, 1.71246089048032811731635317542, 1.89121523431789211349572207683, 2.22716966263648378073160227669, 2.29079055062594220362084331286, 3.06549157842465894591298728496, 3.07573977469608401457911338449, 3.07640697218265104315149821334, 3.87660901936360952201738372395, 3.98017616975734977729022163462, 4.37402390193113619447820536037, 4.62037289882485831277851789444, 4.75672360823448726700323532419, 4.78610251837001875769641906784, 5.43649535240881735650916149932, 5.44596548826748773317403988706, 5.91253508235323922810484612772, 6.35326814101051264697107161697, 6.41799675049240358535866052566, 6.50388505659398930492044929706, 6.50881405915631865574270683295, 6.85157231697169902243924584688, 7.20156858310878440825750243119, 7.34505160909897297023176369342
