| L(s) = 1 | + 4·5-s − 2·9-s + 8·11-s − 12·17-s + 4·19-s − 8·23-s + 10·25-s + 8·27-s + 12·29-s − 24·31-s + 4·37-s + 4·41-s + 12·43-s − 8·45-s − 2·49-s − 28·53-s + 32·55-s + 28·59-s + 4·61-s − 16·71-s − 12·73-s + 8·79-s + 2·81-s − 12·83-s − 48·85-s + 16·95-s + 12·97-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 2/3·9-s + 2.41·11-s − 2.91·17-s + 0.917·19-s − 1.66·23-s + 2·25-s + 1.53·27-s + 2.22·29-s − 4.31·31-s + 0.657·37-s + 0.624·41-s + 1.82·43-s − 1.19·45-s − 2/7·49-s − 3.84·53-s + 4.31·55-s + 3.64·59-s + 0.512·61-s − 1.89·71-s − 1.40·73-s + 0.900·79-s + 2/9·81-s − 1.31·83-s − 5.20·85-s + 1.64·95-s + 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.677109952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.677109952\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} ) \) | 4.3.a_c_ai_c |
| 5 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.5.ae_g_ae_c |
| 7 | $D_4\times C_2$ | \( 1 + 2 T^{2} + 24 T^{3} + 2 T^{4} + 24 p T^{5} + 2 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_c_y_c |
| 11 | $D_4\times C_2$ | \( 1 - 8 T + 34 T^{2} - 128 T^{3} + 450 T^{4} - 128 p T^{5} + 34 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.11.ai_bi_aey_ri |
| 13 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 170 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_ae_a_ago |
| 19 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 20 T^{3} - 146 T^{4} - 20 p T^{5} + 8 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.19.ae_i_au_afq |
| 23 | $D_4\times C_2$ | \( 1 + 8 T + 66 T^{2} + 416 T^{3} + 2306 T^{4} + 416 p T^{5} + 66 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.23.i_co_qa_dks |
| 29 | $D_4\times C_2$ | \( 1 - 12 T + 86 T^{2} - 572 T^{3} + 3362 T^{4} - 572 p T^{5} + 86 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.29.am_di_awa_ezi |
| 31 | $D_4\times C_2$ | \( 1 + 24 T + 242 T^{2} + 1440 T^{3} + 7394 T^{4} + 1440 p T^{5} + 242 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) | 4.31.y_ji_cdk_kyk |
| 37 | $D_4\times C_2$ | \( 1 - 4 T + 22 T^{2} + 316 T^{3} - 1310 T^{4} + 316 p T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.37.ae_w_me_abyk |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 + 6 T + p T^{2} )^{2}( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} ) \) | 4.41.ae_cc_atw_eas |
| 43 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.am_cu_abce_knu |
| 47 | $D_4\times C_2$ | \( 1 - 116 T^{2} + 7270 T^{4} - 116 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_aem_a_ktq |
| 53 | $D_4\times C_2$ | \( 1 + 28 T + 392 T^{2} + 3780 T^{3} + 29726 T^{4} + 3780 p T^{5} + 392 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8} \) | 4.53.bc_pc_fpk_brzi |
| 59 | $D_4\times C_2$ | \( 1 - 28 T + 392 T^{2} - 3948 T^{3} + 33038 T^{4} - 3948 p T^{5} + 392 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) | 4.59.abc_pc_afvw_bwws |
| 61 | $D_4\times C_2$ | \( 1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) | 4.61.ae_g_ae_c |
| 67 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_m_a_nis |
| 71 | $D_4\times C_2$ | \( 1 + 16 T + 82 T^{2} - 200 T^{3} - 4446 T^{4} - 200 p T^{5} + 82 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) | 4.71.q_de_ahs_agpa |
| 73 | $D_4\times C_2$ | \( 1 + 12 T + 38 T^{2} - 1044 T^{3} - 12094 T^{4} - 1044 p T^{5} + 38 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.73.m_bm_aboe_arxe |
| 79 | $D_4\times C_2$ | \( 1 - 8 T + 66 T^{2} + 752 T^{3} - 7038 T^{4} + 752 p T^{5} + 66 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) | 4.79.ai_co_bcy_akks |
| 83 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 1020 T^{3} + 14446 T^{4} + 1020 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.83.m_cu_bng_vjq |
| 89 | $D_4\times C_2$ | \( 1 - 212 T^{2} + 25030 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_aie_a_blas |
| 97 | $D_4\times C_2$ | \( 1 - 12 T + 198 T^{2} - 2652 T^{3} + 28098 T^{4} - 2652 p T^{5} + 198 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) | 4.97.am_hq_adya_bpos |
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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
−8.648124638520558366138143073636, −8.633135536545404642460901462865, −8.440035291347802986216277643384, −7.923294218679069354513000899993, −7.65397142622086832900089085487, −7.05994417689008671884234708187, −7.01332679721364309368773861458, −6.86498068143495589117290522298, −6.64374875640234303703661688705, −6.08733382404629441902701944210, −6.02572957610967292498675849452, −5.88835584256269494730378706485, −5.72576126558341149041430934086, −5.13529385259589767560914338792, −4.65090047454886797299449298009, −4.57803445976502553426210750327, −4.42019641291143566518824535509, −3.72491872733675732278780407206, −3.69599147010651437227079086384, −3.05529727389946765694114540734, −2.76073038950919752249662816146, −2.12324090376699897376538671248, −1.99415673227924278561254766631, −1.62859883846385908683281569708, −0.833752931125255340900151776931,
0.833752931125255340900151776931, 1.62859883846385908683281569708, 1.99415673227924278561254766631, 2.12324090376699897376538671248, 2.76073038950919752249662816146, 3.05529727389946765694114540734, 3.69599147010651437227079086384, 3.72491872733675732278780407206, 4.42019641291143566518824535509, 4.57803445976502553426210750327, 4.65090047454886797299449298009, 5.13529385259589767560914338792, 5.72576126558341149041430934086, 5.88835584256269494730378706485, 6.02572957610967292498675849452, 6.08733382404629441902701944210, 6.64374875640234303703661688705, 6.86498068143495589117290522298, 7.01332679721364309368773861458, 7.05994417689008671884234708187, 7.65397142622086832900089085487, 7.923294218679069354513000899993, 8.440035291347802986216277643384, 8.633135536545404642460901462865, 8.648124638520558366138143073636
