| 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 + ⋯ |
| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.506671962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.506671962\) |
| \(L(\frac{3}{2})\) |
|
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_bk_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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) | 4.23.a_dg_a_eeo |
| 29 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_abc_a_cug |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) | 4.31.a_ca_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 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) | 4.47.a_am_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_aga_a_uag |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.67.a_ki_a_bnvy |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) | 4.71.a_aee_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.i_oq_dfk_cyqw |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.97.ay_xg_alpw_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
−6.76368947674859431460870959687, −6.57422756343468098576802943509, −6.36921521521164189141864332501, −6.22783085235467156227179405390, −5.95885773387392065252218857080, −5.62571351999058326321695828157, −5.50248388980466448592687575236, −5.42765381152160514717305868538, −4.87598968426192542203092021489, −4.86205809218635912770166615541, −4.84747800231023645809864004616, −4.60130616550881524474584273315, −4.05155508992164448297632912867, −3.62633993415349024223016205785, −3.48801615842988575536236348281, −3.28965541515578707849951865382, −2.93088471236026239052555828914, −2.88063971701123431364709843756, −2.73913812502171533258711311120, −2.33216948850350465641759074140, −2.13017551992483252965502554027, −1.40634542141111269094381189358, −1.33788788936343545814428595684, −0.49395299866851651567725660205, −0.42918242374921526987619821287,
0.42918242374921526987619821287, 0.49395299866851651567725660205, 1.33788788936343545814428595684, 1.40634542141111269094381189358, 2.13017551992483252965502554027, 2.33216948850350465641759074140, 2.73913812502171533258711311120, 2.88063971701123431364709843756, 2.93088471236026239052555828914, 3.28965541515578707849951865382, 3.48801615842988575536236348281, 3.62633993415349024223016205785, 4.05155508992164448297632912867, 4.60130616550881524474584273315, 4.84747800231023645809864004616, 4.86205809218635912770166615541, 4.87598968426192542203092021489, 5.42765381152160514717305868538, 5.50248388980466448592687575236, 5.62571351999058326321695828157, 5.95885773387392065252218857080, 6.22783085235467156227179405390, 6.36921521521164189141864332501, 6.57422756343468098576802943509, 6.76368947674859431460870959687
