| L(s) = 1 | + 6·5-s + 5·9-s + 19·25-s + 14·37-s + 30·45-s − 28·49-s − 24·53-s + 9·81-s + 18·89-s + 34·97-s + 42·113-s − 22·121-s + 66·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 52·169-s + 173-s + 179-s + 181-s + 84·185-s + ⋯ |
| L(s) = 1 | + 2.68·5-s + 5/3·9-s + 19/5·25-s + 2.30·37-s + 4.47·45-s − 4·49-s − 3.29·53-s + 81-s + 1.90·89-s + 3.45·97-s + 3.95·113-s − 2·121-s + 5.90·125-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 + 4·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 6.17·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.800030873\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.800030873\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) | |
| good | 3 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} )( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} ) \) | 4.3.a_af_a_q |
| 5 | $C_2^2$ | \( ( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.ag_r_acc_ga |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.7.a_bc_a_li |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.13.a_aca_a_bna |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.17.a_acq_a_cos |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.19.a_cy_a_dfi |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 9 T + 58 T^{2} - 9 p T^{3} + p^{2} T^{4} )( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} ) \) | 4.23.a_bj_a_bau |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.29.a_aem_a_hmc |
| 31 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} ) \) | 4.31.a_abl_a_ps |
| 37 | $C_2^2$ | \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.ao_cv_abak_jqi |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.41.a_agi_a_oxy |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.43.a_gq_a_qks |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) | 4.47.a_adw_a_kgc |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.53.y_qm_gya_ciss |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} )( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} ) \) | 4.59.a_ed_a_lum |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.61.a_ajk_a_bhas |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} )( 1 + 13 T + 102 T^{2} + 13 p T^{3} + p^{2} T^{4} ) \) | 4.67.a_bj_a_aevo |
| 71 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 3 T - 62 T^{2} - 3 p T^{3} + p^{2} T^{4} )( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} ) \) | 4.71.a_afd_a_ssm |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.73.a_alg_a_bvhu |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.79.a_me_a_cdkg |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.83.a_mu_a_cjdu |
| 89 | $C_2^2$ | \( ( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.as_cn_acec_bswi |
| 97 | $C_2^2$ | \( ( 1 - 17 T + 192 T^{2} - 17 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.abi_zx_aony_gjia |
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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
−7.45279336924087024967902506174, −7.37403041010297364130305834221, −6.86274583783224106016723721337, −6.69071688197310778115376908741, −6.57199379865690091315415015121, −6.27378809131363283125655169805, −6.05420586410217754925728117394, −5.93832061135932763866507331135, −5.90617625371826688639988007494, −5.26783565647143304400367488667, −5.03717445339474407680196915613, −4.95024248034381384755391670478, −4.57232690369064062137807117199, −4.52121987168902067654342974922, −4.27448998901315066900756194481, −3.71311839794002161989269748394, −3.25040194358592486475394653805, −3.18481079677874530557982450803, −3.00033106646309711977533788855, −2.33944499646651753526895157690, −2.04251960806278962926392859041, −1.89650285076714457536701544820, −1.60765128419964104500708531684, −1.23755925305209312196101703279, −0.69872148057116350916872204151,
0.69872148057116350916872204151, 1.23755925305209312196101703279, 1.60765128419964104500708531684, 1.89650285076714457536701544820, 2.04251960806278962926392859041, 2.33944499646651753526895157690, 3.00033106646309711977533788855, 3.18481079677874530557982450803, 3.25040194358592486475394653805, 3.71311839794002161989269748394, 4.27448998901315066900756194481, 4.52121987168902067654342974922, 4.57232690369064062137807117199, 4.95024248034381384755391670478, 5.03717445339474407680196915613, 5.26783565647143304400367488667, 5.90617625371826688639988007494, 5.93832061135932763866507331135, 6.05420586410217754925728117394, 6.27378809131363283125655169805, 6.57199379865690091315415015121, 6.69071688197310778115376908741, 6.86274583783224106016723721337, 7.37403041010297364130305834221, 7.45279336924087024967902506174
