| L(s) = 1 | + 4·5-s + 8·9-s + 10·25-s − 8·37-s + 32·45-s − 4·49-s − 24·53-s + 30·81-s + 48·89-s − 8·97-s − 24·113-s − 10·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s − 32·185-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 8/3·9-s + 2·25-s − 1.31·37-s + 4.77·45-s − 4/7·49-s − 3.29·53-s + 10/3·81-s + 5.08·89-s − 0.812·97-s − 2.25·113-s − 0.909·121-s + 1.78·125-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 + 3.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s − 2.35·185-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.439639298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.439639298\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) | |
| good | 3 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.3.a_ai_a_bi |
| 7 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_e_a_dy |
| 13 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_abo_a_bck |
| 17 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_bo_a_blq |
| 19 | $C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_ca_a_cbu |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_adk_a_ele |
| 29 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_acq_a_efe |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.31.a_aeu_a_inu |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.37.i_gq_bjk_ouw |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.41.a_agi_a_oxy |
| 43 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.43.a_fs_a_now |
| 47 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_adk_a_jkk |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.53.y_qm_gya_ciss |
| 59 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) | 4.59.a_aim_a_bcfe |
| 61 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_aca_a_mag |
| 67 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) | 4.67.a_ce_a_olm |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_aim_a_bgve |
| 73 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_ahc_a_bchm |
| 79 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_dw_a_weg |
| 83 | $C_2^2$ | \( ( 1 + 58 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_em_a_zji |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) | 4.89.abw_buy_abdeu_mqmo |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) | 4.97.i_pw_dmu_dmla |
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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
−7.11720722685696995751387718491, −6.89391106960445124531507372891, −6.74343540152821515679999388310, −6.67739698358866920171842339879, −6.53335202427165147795795419822, −6.21336740968476408882762922841, −5.76260173485134442525644345628, −5.67720864879623676247459389023, −5.59212409850118962158581672156, −5.06352411353058574993002668635, −4.84567824441719611062601231832, −4.81275495260217578093530168311, −4.41811681320383772132919493717, −4.31989752042345249206070544106, −4.01530886996188396723436066872, −3.49097719524617578774279015979, −3.28675815834908145579279477773, −3.21773274376224227164490406373, −2.74943591603418109399980349900, −2.13350737286571655270967702605, −2.06930311392595200892033454278, −1.74441559480447174465210878098, −1.53479445132056958053413866036, −1.16574993162747342477669677027, −0.60079907078416439953282864151,
0.60079907078416439953282864151, 1.16574993162747342477669677027, 1.53479445132056958053413866036, 1.74441559480447174465210878098, 2.06930311392595200892033454278, 2.13350737286571655270967702605, 2.74943591603418109399980349900, 3.21773274376224227164490406373, 3.28675815834908145579279477773, 3.49097719524617578774279015979, 4.01530886996188396723436066872, 4.31989752042345249206070544106, 4.41811681320383772132919493717, 4.81275495260217578093530168311, 4.84567824441719611062601231832, 5.06352411353058574993002668635, 5.59212409850118962158581672156, 5.67720864879623676247459389023, 5.76260173485134442525644345628, 6.21336740968476408882762922841, 6.53335202427165147795795419822, 6.67739698358866920171842339879, 6.74343540152821515679999388310, 6.89391106960445124531507372891, 7.11720722685696995751387718491
