| L(s) = 1 | + 16·13-s + 16·25-s − 24·37-s − 4·49-s − 8·61-s − 32·97-s + 48·109-s + 20·121-s + ⋯ |
| L(s) = 1 | + 4.43·13-s + 16/5·25-s − 3.94·37-s − 4/7·49-s − 1.02·61-s − 3.24·97-s + 4.59·109-s + 1.81·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.628221132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.628221132\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_aq_a_ek |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_e_a_dy |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_au_a_ne |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.13.aq_fs_abhw_fow |
| 17 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abg_a_bgc |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.19.a_acy_a_dfi |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_bc_a_bwg |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_aei_a_hdi |
| 31 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_ado_a_fzi |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.37.y_oa_ffs_blso |
| 41 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_bg_a_fje |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_abs_a_gew |
| 47 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_eu_a_mfu |
| 53 | $C_2^2$ | \( ( 1 - 88 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_agu_a_tty |
| 59 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_au_a_klq |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.61.i_ki_cfo_bljy |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_afk_a_unu |
| 71 | $C_2^2$ | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_im_a_bgve |
| 73 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.73.a_lg_a_bvhu |
| 79 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_aky_a_bwhq |
| 83 | $C_2^2$ | \( ( 1 + 134 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_ki_a_buyo |
| 89 | $C_2^2$ | \( ( 1 - 160 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_ami_a_cjhy |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) | 4.97.bg_bds_quy_hruc |
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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
−8.642666308164373347887580215638, −8.405246948675098042165002227671, −8.380563515705680089235661890331, −7.83883647137408232951340544555, −7.72772175526932976034699506303, −7.12286463458675843000321819137, −6.85672940463058260962364787398, −6.75756199129673234147427951280, −6.72782536425731826204413328996, −6.15045363800181020716343317172, −5.91809348584164147656764359401, −5.86877845674063734686819743619, −5.35211201670349974964006866122, −5.19094359376972883526144959599, −4.75008815234871692264760887553, −4.46075670744772068972787034508, −4.16004306840503940575054079219, −3.62885744796318735999549096368, −3.34792848021892115755775602456, −3.34668919802155883136794230939, −3.06009488420297902369055896034, −2.27276346214013939093115467313, −1.61484340692060727880892807346, −1.41187595928568032468506747292, −0.913998941652980752511952537535,
0.913998941652980752511952537535, 1.41187595928568032468506747292, 1.61484340692060727880892807346, 2.27276346214013939093115467313, 3.06009488420297902369055896034, 3.34668919802155883136794230939, 3.34792848021892115755775602456, 3.62885744796318735999549096368, 4.16004306840503940575054079219, 4.46075670744772068972787034508, 4.75008815234871692264760887553, 5.19094359376972883526144959599, 5.35211201670349974964006866122, 5.86877845674063734686819743619, 5.91809348584164147656764359401, 6.15045363800181020716343317172, 6.72782536425731826204413328996, 6.75756199129673234147427951280, 6.85672940463058260962364787398, 7.12286463458675843000321819137, 7.72772175526932976034699506303, 7.83883647137408232951340544555, 8.380563515705680089235661890331, 8.405246948675098042165002227671, 8.642666308164373347887580215638
