| L(s) = 1 | + 4·4-s + 8·11-s + 12·16-s − 8·25-s + 32·44-s − 24·49-s − 16·59-s + 32·64-s + 8·67-s − 56·89-s + 64·97-s − 32·100-s + 24·113-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 16·169-s + 173-s + 96·176-s + 179-s + ⋯ |
| L(s) = 1 | + 2·4-s + 2.41·11-s + 3·16-s − 8/5·25-s + 4.82·44-s − 3.42·49-s − 2.08·59-s + 4·64-s + 0.977·67-s − 5.93·89-s + 6.49·97-s − 3.19·100-s + 2.25·113-s + 2.36·121-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 − 1.23·169-s + 0.0760·173-s + 7.23·176-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.266522447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.266522447\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) | |
| 3 | | \( 1 \) | |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_i_a_co |
| 7 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) | 4.7.a_y_a_ji |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_q_a_pm |
| 17 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.17.a_am_a_xq |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) | 4.19.a_au_a_bfq |
| 23 | $C_2^2$ | \( ( 1 - 32 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_acm_a_dcc |
| 29 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_dw_a_gew |
| 31 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_am_a_cxi |
| 37 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_abk_a_enu |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.41.a_agi_a_oxy |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) | 4.43.a_aem_a_klq |
| 47 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_age_a_qac |
| 53 | $C_2^2$ | \( ( 1 - 92 T^{2} + p^{2} T^{4} )^{2} \) | 4.53.a_ahc_a_uvq |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) | 4.59.q_mu_eou_bwaw |
| 61 | $C_2^2$ | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{2} \) | 4.61.a_adc_a_nju |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.67.ai_lg_aclc_bsqg |
| 71 | $C_2^2$ | \( ( 1 - 128 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_ajw_a_bndy |
| 73 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_acq_a_rmk |
| 79 | $C_2^2$ | \( ( 1 + 156 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_ma_a_ccmc |
| 83 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) | 4.83.a_aee_a_ysc |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{4} \) | 4.89.ce_cgy_bmjg_quuo |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{4} \) | 4.97.acm_cwa_abzum_xxgw |
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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
−7.39345989259169430989193317898, −7.10845237010199582380031142428, −6.87564013572564711974514318646, −6.57685949983330472684934266203, −6.50621820237977268712519485808, −6.32663460191019919016830606683, −6.05518702572354481769245134552, −5.93232229476888738874276300817, −5.72724274737063986707503828777, −5.32168677338338074402047571622, −5.05256604158683674112416347453, −4.74246395921571450634281757166, −4.49035187181517343005132029721, −4.15050781063747812875066670923, −3.82871688041778178366007746446, −3.70809373498791267855407704611, −3.35679792210623701549396367394, −3.02975774761631823862746576705, −2.99502312136803680802879816672, −2.45559805005103225557827116992, −1.99382457730382349007250320704, −1.69915456181324315921388795798, −1.65042045513201266856914161166, −1.26838794116559262311026992333, −0.58321160517904228242432480189,
0.58321160517904228242432480189, 1.26838794116559262311026992333, 1.65042045513201266856914161166, 1.69915456181324315921388795798, 1.99382457730382349007250320704, 2.45559805005103225557827116992, 2.99502312136803680802879816672, 3.02975774761631823862746576705, 3.35679792210623701549396367394, 3.70809373498791267855407704611, 3.82871688041778178366007746446, 4.15050781063747812875066670923, 4.49035187181517343005132029721, 4.74246395921571450634281757166, 5.05256604158683674112416347453, 5.32168677338338074402047571622, 5.72724274737063986707503828777, 5.93232229476888738874276300817, 6.05518702572354481769245134552, 6.32663460191019919016830606683, 6.50621820237977268712519485808, 6.57685949983330472684934266203, 6.87564013572564711974514318646, 7.10845237010199582380031142428, 7.39345989259169430989193317898
