| L(s) = 1 | + 4·2-s − 8·3-s + 8·4-s − 32·6-s + 8·8-s + 28·9-s − 64·12-s − 4·16-s + 112·18-s − 64·24-s + 8·25-s − 40·27-s − 32·32-s + 224·36-s − 16·41-s + 32·48-s − 28·49-s + 32·50-s − 160·54-s − 8·59-s − 64·64-s + 224·72-s + 32·73-s − 64·75-s − 70·81-s − 64·82-s + 256·96-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 4.61·3-s + 4·4-s − 13.0·6-s + 2.82·8-s + 28/3·9-s − 18.4·12-s − 16-s + 26.3·18-s − 13.0·24-s + 8/5·25-s − 7.69·27-s − 5.65·32-s + 37.3·36-s − 2.49·41-s + 4.61·48-s − 4·49-s + 4.52·50-s − 21.7·54-s − 1.04·59-s − 8·64-s + 26.3·72-s + 3.74·73-s − 7.39·75-s − 7.77·81-s − 7.06·82-s + 26.1·96-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4062830395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4062830395\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) | |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) | |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.3.i_bk_ea_ig |
| 5 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_ai_a_co |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.7.a_bc_a_li |
| 11 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.11.a_aq_a_lu |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) | 4.13.a_au_a_qw |
| 17 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_bs_a_bow |
| 19 | $C_2^2$ | \( ( 1 - 24 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_abw_a_bxy |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.29.a_aem_a_hmc |
| 31 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_aem_a_hvi |
| 37 | $C_2^2$ | \( ( 1 + 60 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_eq_a_jju |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.41.q_ka_dho_bayo |
| 43 | $C_2^2$ | \( ( 1 - 72 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_afo_a_ndq |
| 47 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_aga_a_pny |
| 53 | $C_2^2$ | \( ( 1 + 92 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_hc_a_uvq |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.59.i_ka_cds_bjcw |
| 61 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ai_a_law |
| 67 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_aq_a_nju |
| 71 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_e_a_oxy |
| 73 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.73.abg_baa_anki_fghq |
| 79 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_hw_a_bhwg |
| 83 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_adc_a_wtm |
| 89 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_ajk_a_btlu |
| 97 | $C_2^2$ | \( ( 1 - 138 T^{2} + p^{2} T^{4} )^{2} \) | 4.97.a_akq_a_ceag |
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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
−9.450297432522309127845118616495, −8.989207658483847820091447243225, −8.608616022707566008882485854245, −8.556380289018904717912945690740, −7.962113268397208231228109764394, −7.73605897206735795952526287179, −6.93300420890005330966792667750, −6.74261574764646118075437356507, −6.60643130094261070477138724405, −6.59858426295856719502073322758, −6.32854345382886087974174240767, −5.87674156205407274372902264969, −5.85371545182580844712772491264, −5.25953054986508603221972794454, −5.21671665331048608015391716302, −5.17701535225859987080360240440, −4.83031192695072343460184636443, −4.54888162832260467836773133349, −4.43880681327672301792311917011, −3.63973179647490228840212486579, −3.23924115048015844390918395220, −3.09822028792829028321770098491, −2.42136480627944769424890643112, −1.43921759295853468101761526996, −0.36539487028182264579499464005,
0.36539487028182264579499464005, 1.43921759295853468101761526996, 2.42136480627944769424890643112, 3.09822028792829028321770098491, 3.23924115048015844390918395220, 3.63973179647490228840212486579, 4.43880681327672301792311917011, 4.54888162832260467836773133349, 4.83031192695072343460184636443, 5.17701535225859987080360240440, 5.21671665331048608015391716302, 5.25953054986508603221972794454, 5.85371545182580844712772491264, 5.87674156205407274372902264969, 6.32854345382886087974174240767, 6.59858426295856719502073322758, 6.60643130094261070477138724405, 6.74261574764646118075437356507, 6.93300420890005330966792667750, 7.73605897206735795952526287179, 7.962113268397208231228109764394, 8.556380289018904717912945690740, 8.608616022707566008882485854245, 8.989207658483847820091447243225, 9.450297432522309127845118616495
