| L(s) = 1 | + 8·7-s − 4·13-s − 16-s − 16·31-s − 4·37-s + 32·43-s + 32·49-s + 32·61-s − 16·67-s − 4·73-s − 32·91-s + 44·97-s − 40·103-s − 8·112-s + 28·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 1.10·13-s − 1/4·16-s − 2.87·31-s − 0.657·37-s + 4.87·43-s + 32/7·49-s + 4.09·61-s − 1.95·67-s − 0.468·73-s − 3.35·91-s + 4.46·97-s − 3.94·103-s − 0.755·112-s + 2.54·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 8/13·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.300646045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.300646045\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | | \( 1 \) | |
| 5 | | \( 1 \) | |
| good | 2 | $C_2^3$ | \( 1 + T^{4} + p^{4} T^{8} \) | 4.2.a_a_a_b |
| 7 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.ai_bg_aeq_ow |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) | 4.11.a_abc_a_qw |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) | 4.13.e_i_ci_re |
| 17 | $C_2^3$ | \( 1 - 254 T^{4} + p^{4} T^{8} \) | 4.17.a_a_a_aju |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.19.a_acy_a_dfi |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) | 4.23.a_a_a_agc |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_dc_a_ewg |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.31.q_im_cpc_rso |
| 37 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.e_i_ga_emw |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_age_a_olm |
| 43 | $C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.abg_ts_aicm_ckhe |
| 47 | $C_2^3$ | \( 1 - 3518 T^{4} + p^{4} T^{8} \) | 4.47.a_a_a_affi |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )( 1 + 56 T^{2} + p^{2} T^{4} ) \) | 4.53.a_a_a_drm |
| 59 | $C_2^2$ | \( ( 1 + 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_do_a_nle |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.61.abg_ye_alsa_eekc |
| 67 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.q_ey_ciy_bbmo |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_aim_a_bgve |
| 73 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.e_i_lo_qqo |
| 79 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_abc_a_stq |
| 83 | $C_2^3$ | \( 1 + 8722 T^{4} + p^{4} T^{8} \) | 4.83.a_a_a_mxm |
| 89 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_bg_a_xve |
| 97 | $C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.abs_blg_awbs_jtjq |
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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
−9.087039964230170588001187478423, −8.646272972011338520076968077525, −8.299540793805372021990532734293, −8.207357606127557863819765877889, −7.87402606288572129739644673693, −7.60199167602809397153443426663, −7.41439774919426906661234688227, −7.22408914352540503292374627314, −6.95251237299363432805196377715, −6.74815821322440349680070626144, −5.94259361664127967526430284693, −5.77736108683814380360010462353, −5.63791021885016944402708361816, −5.36123438529372113779581079301, −4.92528642053359984071928104512, −4.76327402091490315405806228426, −4.49273148215298350663438184775, −4.01941594741982611705807245109, −3.92599680435465843769319585591, −3.43575388767802316861762507108, −2.67939316892579796084197521046, −2.25848312992364474183984224563, −2.15141207381034754334356551336, −1.60237081858699516247308595620, −0.941842629029834829276222122057,
0.941842629029834829276222122057, 1.60237081858699516247308595620, 2.15141207381034754334356551336, 2.25848312992364474183984224563, 2.67939316892579796084197521046, 3.43575388767802316861762507108, 3.92599680435465843769319585591, 4.01941594741982611705807245109, 4.49273148215298350663438184775, 4.76327402091490315405806228426, 4.92528642053359984071928104512, 5.36123438529372113779581079301, 5.63791021885016944402708361816, 5.77736108683814380360010462353, 5.94259361664127967526430284693, 6.74815821322440349680070626144, 6.95251237299363432805196377715, 7.22408914352540503292374627314, 7.41439774919426906661234688227, 7.60199167602809397153443426663, 7.87402606288572129739644673693, 8.207357606127557863819765877889, 8.299540793805372021990532734293, 8.646272972011338520076968077525, 9.087039964230170588001187478423
