| L(s) = 1 | + 12·25-s + 24·41-s − 12·49-s + 48·73-s − 48·89-s − 32·97-s − 24·113-s + 8·121-s + ⋯ |
| L(s) = 1 | + 12/5·25-s + 3.74·41-s − 1.71·49-s + 5.61·73-s − 5.08·89-s − 3.24·97-s − 2.25·113-s + 8/11·121-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
| \(L(1)\) |
\(\approx\) |
\(1.645239029\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.645239029\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.5.a_am_a_di |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_m_a_fe |
| 11 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_ai_a_jy |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) | 4.13.a_u_a_qw |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.17.a_cq_a_cos |
| 19 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_abo_a_bre |
| 23 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_aca_a_cos |
| 29 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_aee_a_guw |
| 31 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_ci_a_eeo |
| 37 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_acy_a_gew |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.41.ay_oq_afqu_brba |
| 43 | $C_2^2$ | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_afg_a_mic |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.47.a_hg_a_tpu |
| 53 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ahw_a_xsg |
| 59 | $C_2^2$ | \( ( 1 - 116 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_aiy_a_befi |
| 61 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_agq_a_vys |
| 67 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_ce_a_olm |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_fk_a_weg |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) | 4.73.abw_bsm_azue_kepa |
| 79 | $C_2^2$ | \( ( 1 + 126 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_js_a_bpys |
| 83 | $C_2^2$ | \( ( 1 - 148 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_alk_a_cauk |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) | 4.89.bw_buy_bdeu_mqmo |
| 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
−5.69766660488799095355306027795, −5.49464843411884541648503991524, −5.48925330882063606260667882188, −5.41665601052717494459618463585, −5.20698555174115484663407776158, −4.85828853763216057957436404441, −4.62278038144966703305080548027, −4.46349066196804741041833115996, −4.40383905825763872860261312234, −4.02561579042196929689118760266, −3.87856573644499941642861350466, −3.80040219345842476548478547411, −3.47237652716124222915344899352, −3.11386708597702397651688646169, −2.90131600711281574628088162613, −2.78931845186483943953777142930, −2.69964990107574924773716200604, −2.23996783992708590703017263561, −2.22829509403370177921618677229, −1.83277615948988858093719737006, −1.33076500651742457136796478816, −1.27919503700032776849977338528, −0.949714198974464612693349461534, −0.75852322807301730629498627603, −0.17087612944078871439322341313,
0.17087612944078871439322341313, 0.75852322807301730629498627603, 0.949714198974464612693349461534, 1.27919503700032776849977338528, 1.33076500651742457136796478816, 1.83277615948988858093719737006, 2.22829509403370177921618677229, 2.23996783992708590703017263561, 2.69964990107574924773716200604, 2.78931845186483943953777142930, 2.90131600711281574628088162613, 3.11386708597702397651688646169, 3.47237652716124222915344899352, 3.80040219345842476548478547411, 3.87856573644499941642861350466, 4.02561579042196929689118760266, 4.40383905825763872860261312234, 4.46349066196804741041833115996, 4.62278038144966703305080548027, 4.85828853763216057957436404441, 5.20698555174115484663407776158, 5.41665601052717494459618463585, 5.48925330882063606260667882188, 5.49464843411884541648503991524, 5.69766660488799095355306027795
