| L(s) = 1 | − 4·3-s + 4·7-s + 10·9-s − 8·11-s − 16·21-s + 6·25-s − 20·27-s + 32·33-s + 8·41-s + 16·47-s − 8·49-s + 8·53-s + 40·63-s − 4·67-s + 4·73-s − 24·75-s − 32·77-s + 35·81-s + 24·83-s − 80·99-s + 16·101-s − 8·107-s + 36·121-s − 32·123-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 1.51·7-s + 10/3·9-s − 2.41·11-s − 3.49·21-s + 6/5·25-s − 3.84·27-s + 5.57·33-s + 1.24·41-s + 2.33·47-s − 8/7·49-s + 1.09·53-s + 5.03·63-s − 0.488·67-s + 0.468·73-s − 2.77·75-s − 3.64·77-s + 35/9·81-s + 2.63·83-s − 8.04·99-s + 1.59·101-s − 0.773·107-s + 3.27·121-s − 2.88·123-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.060360515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060360515\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) | |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) | |
| good | 5 | $D_4\times C_2$ | \( 1 - 6 T^{2} + 14 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) | 4.5.a_ag_a_o |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 10 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.7.ae_y_acq_ju |
| 11 | $C_4$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.i_bc_fg_yg |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 454 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) | 4.13.a_abc_a_rm |
| 17 | $D_4\times C_2$ | \( 1 - 14 T^{2} + 382 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_ao_a_os |
| 19 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1318 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_aca_a_bys |
| 23 | $D_4\times C_2$ | \( 1 + 34 T^{2} + 1222 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_bi_a_bva |
| 29 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 1502 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) | 4.29.a_abe_a_cfu |
| 31 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_abs_a_doo |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.41.ai_hg_abnc_rwg |
| 43 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_aga_a_omg |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) | 4.47.aq_ky_adsq_bhis |
| 53 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.ai_cy_azo_mdy |
| 59 | $D_4\times C_2$ | \( 1 - 230 T^{2} + 20182 T^{4} - 230 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_aiw_a_bdwg |
| 61 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_agy_a_wzu |
| 67 | $D_{4}$ | \( ( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.e_hc_ye_babm |
| 71 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_eu_a_upq |
| 73 | $D_{4}$ | \( ( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.ae_lc_abhc_buly |
| 79 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 6198 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_bc_a_jek |
| 83 | $D_{4}$ | \( ( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.ay_to_ajkq_eati |
| 89 | $D_4\times C_2$ | \( 1 - 230 T^{2} + 28942 T^{4} - 230 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_aiw_a_bqve |
| 97 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 19734 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) | 4.97.a_agq_a_bdfa |
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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
−7.943292877590901717091025525057, −7.84278729022676306832511203184, −7.48875327155824554270782721120, −7.29850769993290821048598606178, −7.08412817060963723399664858011, −6.69309701469450201278367063601, −6.65258134323036059923835904525, −6.22526793826310887430262328071, −5.76411427351526218026164123328, −5.72459682183535992472364605042, −5.60720483243913277086941191967, −5.24761690332822063716149584052, −5.06406068454349665520706912871, −4.78458778129742216141964141900, −4.46177293159658183100883153492, −4.45705201642717063481790554703, −4.14919994336058509881377221853, −3.39839459191073237504345175532, −3.29315263553553006769799925086, −2.76417633886699127830832873462, −2.25518090008552879901056756309, −2.06526684004951562104115951224, −1.59865815106246346437730187120, −0.74837225273034262343388348330, −0.66836703962159163295753653542,
0.66836703962159163295753653542, 0.74837225273034262343388348330, 1.59865815106246346437730187120, 2.06526684004951562104115951224, 2.25518090008552879901056756309, 2.76417633886699127830832873462, 3.29315263553553006769799925086, 3.39839459191073237504345175532, 4.14919994336058509881377221853, 4.45705201642717063481790554703, 4.46177293159658183100883153492, 4.78458778129742216141964141900, 5.06406068454349665520706912871, 5.24761690332822063716149584052, 5.60720483243913277086941191967, 5.72459682183535992472364605042, 5.76411427351526218026164123328, 6.22526793826310887430262328071, 6.65258134323036059923835904525, 6.69309701469450201278367063601, 7.08412817060963723399664858011, 7.29850769993290821048598606178, 7.48875327155824554270782721120, 7.84278729022676306832511203184, 7.943292877590901717091025525057
