| L(s) = 1 | + 24·23-s − 20·25-s + 24·47-s − 8·49-s − 24·71-s − 24·73-s − 48·97-s + 12·121-s + ⋯ |
| L(s) = 1 | + 5.00·23-s − 4·25-s + 3.50·47-s − 8/7·49-s − 2.84·71-s − 2.80·73-s − 4.87·97-s + 1.09·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.829813608\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.829813608\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.5.a_u_a_fu |
| 7 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_i_a_ek |
| 11 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_am_a_ks |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.13.a_u_a_qw |
| 17 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_abg_a_bgc |
| 19 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ci_a_ckk |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.23.ay_lw_adsy_vic |
| 29 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_abc_a_cug |
| 31 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_adk_a_fsk |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_acy_a_gew |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_age_a_olm |
| 43 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_ga_a_omg |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.47.ay_po_aghk_bzoo |
| 53 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_cq_a_kao |
| 59 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ahw_a_zry |
| 61 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_agq_a_vys |
| 67 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_m_a_nis |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.71.y_tg_ivw_doaw |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.73.y_to_jbk_drwo |
| 79 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_aku_a_bvly |
| 83 | $C_2^2$ | \( ( 1 + 90 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_gy_a_bgjm |
| 89 | $C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_abg_a_xve |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) | 4.97.bw_bwe_bexc_nydq |
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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
−5.78491311749847362212961810698, −5.71596944623108960803519351747, −5.67411040927086991868436406844, −5.26837242631657484972213797180, −5.08522473669796255826782494454, −4.90144293572366410068821107480, −4.79098332872194484303009462565, −4.25940407479232988917759527689, −4.25083217142396390543041661476, −4.25032046289159222792044172277, −3.97914144838994492148450285394, −3.73466003330507951165631283969, −3.25878743773318187017375951468, −3.14405262904506503251786177068, −3.09701847766210319159419498885, −2.89201637154261475087014709040, −2.52040323275803433634820376722, −2.34006616389169269404761624591, −2.18563953736504578282034994336, −1.59980489938982102839267121338, −1.55062399110808090391550943429, −1.28727361565878232570871578636, −1.04728281394661421644117801499, −0.58395077923372593948786094163, −0.20452694834268115496776095470,
0.20452694834268115496776095470, 0.58395077923372593948786094163, 1.04728281394661421644117801499, 1.28727361565878232570871578636, 1.55062399110808090391550943429, 1.59980489938982102839267121338, 2.18563953736504578282034994336, 2.34006616389169269404761624591, 2.52040323275803433634820376722, 2.89201637154261475087014709040, 3.09701847766210319159419498885, 3.14405262904506503251786177068, 3.25878743773318187017375951468, 3.73466003330507951165631283969, 3.97914144838994492148450285394, 4.25032046289159222792044172277, 4.25083217142396390543041661476, 4.25940407479232988917759527689, 4.79098332872194484303009462565, 4.90144293572366410068821107480, 5.08522473669796255826782494454, 5.26837242631657484972213797180, 5.67411040927086991868436406844, 5.71596944623108960803519351747, 5.78491311749847362212961810698
