| L(s) = 1 | − 4·4-s + 4·16-s + 24·31-s + 20·37-s − 18·49-s + 16·64-s + 36·67-s − 4·97-s − 20·103-s − 96·124-s + 127-s + 131-s + 137-s + 139-s − 80·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 72·196-s + ⋯ |
| L(s) = 1 | − 2·4-s + 16-s + 4.31·31-s + 3.28·37-s − 2.57·49-s + 2·64-s + 4.39·67-s − 0.406·97-s − 1.97·103-s − 8.62·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6.57·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 36/7·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.154915598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.154915598\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | | \( 1 \) | |
| good | 2 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) | 4.2.a_e_a_m |
| 5 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.5.a_a_a_by |
| 7 | $C_2^2$ | \( ( 1 + 9 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_s_a_gx |
| 13 | $C_2^2$ | \( ( 1 - 19 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_abm_a_bax |
| 17 | $C_2^2$ | \( ( 1 + 32 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_cm_a_cjq |
| 19 | $C_2^2$ | \( ( 1 + 33 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_co_a_crr |
| 23 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_m_a_bqc |
| 29 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_q_a_cpe |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.31.ay_nc_aepc_begs |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) | 4.37.au_lm_aeaq_bdmx |
| 41 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_ado_a_ics |
| 43 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_m_a_fnq |
| 47 | $C_2^2$ | \( ( 1 + 84 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_gm_a_qzi |
| 53 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_bg_a_iry |
| 59 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_ce_a_lly |
| 61 | $C_2^2$ | \( ( 1 + 117 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ja_a_bfgt |
| 67 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{4} \) | 4.67.abk_bda_apam_fpxb |
| 71 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_hw_a_behy |
| 73 | $C_2^2$ | \( ( 1 + 141 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_kw_a_btep |
| 79 | $C_2^2$ | \( ( 1 + 113 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_is_a_bljf |
| 83 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_me_a_cfic |
| 89 | $C_2^2$ | \( ( 1 - 72 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_afo_a_bfcs |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) | 4.97.e_pe_bsy_dhgd |
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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
−6.24238692348669933535907887088, −5.80659879459319595399223796082, −5.57248855759411087736275143384, −5.54615504965930215617215972546, −5.28593498869453266634596506767, −4.89776964037827872049556107134, −4.89278427099555637520253215282, −4.64762450098929814314285181734, −4.62736986222419266073329878891, −4.14107297828572661121600794150, −4.08518325994074505977306555187, −4.01514464895319818348366095811, −3.99595432959487515373310928613, −3.17834510539579526293263817840, −3.15964860108275395895967084700, −3.11383708092799170440532722229, −2.67644534204805550532167851321, −2.53906647398094881232174099789, −2.13892293525057466707125187448, −2.05715144885827227906772747891, −1.50447947765751282777951999907, −1.18995042772832678603823338573, −0.75918319698188173452632165107, −0.63370616457687553249853153004, −0.45978197053089263039075990729,
0.45978197053089263039075990729, 0.63370616457687553249853153004, 0.75918319698188173452632165107, 1.18995042772832678603823338573, 1.50447947765751282777951999907, 2.05715144885827227906772747891, 2.13892293525057466707125187448, 2.53906647398094881232174099789, 2.67644534204805550532167851321, 3.11383708092799170440532722229, 3.15964860108275395895967084700, 3.17834510539579526293263817840, 3.99595432959487515373310928613, 4.01514464895319818348366095811, 4.08518325994074505977306555187, 4.14107297828572661121600794150, 4.62736986222419266073329878891, 4.64762450098929814314285181734, 4.89278427099555637520253215282, 4.89776964037827872049556107134, 5.28593498869453266634596506767, 5.54615504965930215617215972546, 5.57248855759411087736275143384, 5.80659879459319595399223796082, 6.24238692348669933535907887088
