| L(s) = 1 | + 4·3-s + 8·9-s − 12·11-s − 16-s − 12·23-s + 20·27-s + 8·31-s − 48·33-s + 4·37-s + 4·47-s − 4·48-s − 12·53-s − 12·67-s − 48·69-s + 32·71-s + 50·81-s + 32·93-s + 28·97-s − 96·99-s + 12·103-s + 16·111-s − 4·113-s + 86·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s − 3.61·11-s − 1/4·16-s − 2.50·23-s + 3.84·27-s + 1.43·31-s − 8.35·33-s + 0.657·37-s + 0.583·47-s − 0.577·48-s − 1.64·53-s − 1.46·67-s − 5.77·69-s + 3.79·71-s + 50/9·81-s + 3.31·93-s + 2.84·97-s − 9.64·99-s + 1.18·103-s + 1.51·111-s − 0.376·113-s + 7.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.970630245\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.970630245\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) | |
| 5 | | \( 1 \) | |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) | |
| good | 3 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.3.ae_i_au_bu |
| 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) | 4.7.a_a_a_adq |
| 13 | $C_2^3$ | \( 1 + 146 T^{4} + p^{4} T^{8} \) | 4.13.a_a_a_fq |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T^{2} + p^{2} T^{4} )( 1 + 16 T^{2} + p^{2} T^{4} ) \) | 4.17.a_a_a_mk |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_acq_a_cug |
| 23 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.23.m_cu_sy_emw |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_dw_a_gew |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.31.ai_fs_abdw_ktq |
| 37 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ae_i_aga_emw |
| 41 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_au_a_fde |
| 43 | $C_2^3$ | \( 1 + 1202 T^{4} + p^{4} T^{8} \) | 4.43.a_a_a_bug |
| 47 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.ae_i_aho_hco |
| 53 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.m_cu_bgu_oli |
| 59 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_abk_a_kug |
| 61 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_ga_a_uag |
| 67 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.m_cu_bng_uxi |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) | 4.71.abg_zs_ancy_fbmc |
| 73 | $C_2^3$ | \( 1 - 8542 T^{4} + p^{4} T^{8} \) | 4.73.a_a_a_amqo |
| 79 | $C_2^2$ | \( ( 1 + 150 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_lo_a_bztm |
| 83 | $C_2^3$ | \( 1 - 3374 T^{4} + p^{4} T^{8} \) | 4.83.a_a_a_aezu |
| 89 | $C_2^2$ | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_aky_a_cbgw |
| 97 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.abc_pc_aica_duhq |
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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
−7.71082394977352466545176247777, −7.66366110644132533184844970815, −7.62017962620729214141515940863, −7.36498870745151934871850648807, −6.94181453631629259622781549775, −6.59550377988429378278499523272, −6.38788364695805624321318583858, −6.02925844404980545312955001098, −5.93762767602204364548489402459, −5.61836882815009545707689232108, −5.17521114688469297770740333753, −4.86777193707018690641613167895, −4.70841628464075650260613655322, −4.67770975293161068727991561326, −4.26049608900297004599547513572, −3.62825607533518697286864775959, −3.60424085191885521993802342908, −3.29990984621178309365733394210, −2.91876818002377319181911031941, −2.57277604801762574354164224214, −2.45552148977766131604290559548, −2.24618028413130574557880861280, −2.00520936223698616739704904486, −1.22510892952274834074853674651, −0.43893739910340033527428080513,
0.43893739910340033527428080513, 1.22510892952274834074853674651, 2.00520936223698616739704904486, 2.24618028413130574557880861280, 2.45552148977766131604290559548, 2.57277604801762574354164224214, 2.91876818002377319181911031941, 3.29990984621178309365733394210, 3.60424085191885521993802342908, 3.62825607533518697286864775959, 4.26049608900297004599547513572, 4.67770975293161068727991561326, 4.70841628464075650260613655322, 4.86777193707018690641613167895, 5.17521114688469297770740333753, 5.61836882815009545707689232108, 5.93762767602204364548489402459, 6.02925844404980545312955001098, 6.38788364695805624321318583858, 6.59550377988429378278499523272, 6.94181453631629259622781549775, 7.36498870745151934871850648807, 7.62017962620729214141515940863, 7.66366110644132533184844970815, 7.71082394977352466545176247777
