| L(s) = 1 | + 4·3-s + 8·9-s − 4·11-s + 18·25-s + 12·27-s − 16·33-s − 4·47-s + 22·49-s − 32·59-s + 12·61-s + 32·71-s − 52·73-s + 72·75-s + 23·81-s − 24·83-s + 32·97-s − 32·99-s − 24·107-s − 24·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 16·141-s + 88·147-s + 149-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 8/3·9-s − 1.20·11-s + 18/5·25-s + 2.30·27-s − 2.78·33-s − 0.583·47-s + 22/7·49-s − 4.16·59-s + 1.53·61-s + 3.79·71-s − 6.08·73-s + 8.31·75-s + 23/9·81-s − 2.63·83-s + 3.24·97-s − 3.21·99-s − 2.32·107-s − 2.29·109-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.34·141-s + 7.25·147-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.542667678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.542667678\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) | |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) | |
| good | 5 | $C_2^2$ | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{2} \) | 4.5.a_as_a_fb |
| 7 | $D_4\times C_2$ | \( 1 - 22 T^{2} + 211 T^{4} - 22 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_aw_a_id |
| 11 | $D_{4}$ | \( ( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.11.e_o_cm_nr |
| 13 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) | 4.13.a_q_a_pm |
| 17 | $D_4\times C_2$ | \( 1 - 50 T^{2} + 1171 T^{4} - 50 p^{2} T^{6} + p^{4} T^{8} \) | 4.17.a_aby_a_btb |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_bc_a_bwg |
| 29 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_m_a_coc |
| 31 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1554 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_ace_a_chu |
| 37 | $C_2^2$ | \( ( 1 + 24 T^{2} + p^{2} T^{4} )^{2} \) | 4.37.a_bw_a_exm |
| 41 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 3106 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) | 4.41.a_abg_a_epm |
| 43 | $D_4\times C_2$ | \( 1 - 38 T^{2} + 531 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_abm_a_ul |
| 47 | $D_{4}$ | \( ( 1 + 2 T + 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.47.e_hi_vo_txb |
| 53 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 7522 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) | 4.53.a_aea_a_ldi |
| 59 | $D_{4}$ | \( ( 1 + 16 T + 164 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.59.bg_wm_kom_dquc |
| 61 | $D_{4}$ | \( ( 1 - 6 T + 99 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.am_ja_acvw_bgad |
| 67 | $D_4\times C_2$ | \( 1 - 180 T^{2} + 15926 T^{4} - 180 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_agy_a_xoo |
| 71 | $D_{4}$ | \( ( 1 - 16 T + 204 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.71.abg_zo_anam_fago |
| 73 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) | 4.73.ca_byg_bdwa_lwoh |
| 79 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 24326 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_aiu_a_bjzq |
| 83 | $D_{4}$ | \( ( 1 + 12 T + 170 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.83.y_sq_izo_dumw |
| 89 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 12242 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) | 4.89.a_aeq_a_scw |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 240 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.97.abg_bci_apyu_heni |
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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
−6.75335276133197860871103461271, −6.28182550649008301212692236433, −6.13651718596831791617031309417, −6.03975573637421776187088789014, −5.91831002661877279876437815541, −5.37423491431659611001902943156, −5.15925989881242844883514285128, −4.98243689935058668842042429288, −4.94709945516247307600744921976, −4.56433944219229557881131821116, −4.46203167293979328490776112491, −3.98102652443344655071644524944, −3.89511724715643821139402727171, −3.75913221520507921973222235186, −3.18971741704475985622753291326, −3.10864021902294305569378499055, −2.89971825999687955985932842381, −2.68971550861647663543083942205, −2.65143190217328971815304251230, −2.34147347762305015824278951200, −1.92384986377767600100990660468, −1.54851952011301848774886412835, −1.29860335700191568469328401431, −0.946003330300619672429969644146, −0.38087384084979179847915273809,
0.38087384084979179847915273809, 0.946003330300619672429969644146, 1.29860335700191568469328401431, 1.54851952011301848774886412835, 1.92384986377767600100990660468, 2.34147347762305015824278951200, 2.65143190217328971815304251230, 2.68971550861647663543083942205, 2.89971825999687955985932842381, 3.10864021902294305569378499055, 3.18971741704475985622753291326, 3.75913221520507921973222235186, 3.89511724715643821139402727171, 3.98102652443344655071644524944, 4.46203167293979328490776112491, 4.56433944219229557881131821116, 4.94709945516247307600744921976, 4.98243689935058668842042429288, 5.15925989881242844883514285128, 5.37423491431659611001902943156, 5.91831002661877279876437815541, 6.03975573637421776187088789014, 6.13651718596831791617031309417, 6.28182550649008301212692236433, 6.75335276133197860871103461271
