| L(s) = 1 | + 2·5-s + 9-s + 4·13-s + 10·17-s + 3·25-s + 8·29-s − 14·37-s + 20·41-s + 2·45-s − 21·49-s − 20·53-s + 8·65-s + 4·73-s − 7·81-s + 20·85-s + 20·89-s + 40·97-s − 24·101-s + 18·109-s + 20·113-s + 4·117-s − 30·121-s + 14·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1/3·9-s + 1.10·13-s + 2.42·17-s + 3/5·25-s + 1.48·29-s − 2.30·37-s + 3.12·41-s + 0.298·45-s − 3·49-s − 2.74·53-s + 0.992·65-s + 0.468·73-s − 7/9·81-s + 2.16·85-s + 2.11·89-s + 4.06·97-s − 2.38·101-s + 1.72·109-s + 1.88·113-s + 0.369·117-s − 2.72·121-s + 1.25·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.33924661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.33924661\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 3 | $C_2^2 \wr C_2$ | \( 1 - T^{2} + 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) | 4.3.a_ab_a_i |
| 5 | $D_{4}$ | \( ( 1 - T - p T^{3} + p^{2} T^{4} )^{2} \) | 4.5.ac_b_ak_ci |
| 7 | $C_2^2 \wr C_2$ | \( 1 + 3 p T^{2} + 198 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8} \) | 4.7.a_v_a_hq |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 30 T^{2} + 426 T^{4} + 30 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_be_a_qk |
| 17 | $D_{4}$ | \( ( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) | 4.17.ak_dh_asc_dlo |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 50 T^{2} + 1306 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_by_a_byg |
| 23 | $C_2^2 \wr C_2$ | \( 1 + 40 T^{2} + 1294 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_bo_a_bxu |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) | 4.29.ai_fk_abca_jog |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 66 T^{2} + 2970 T^{4} + 66 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_co_a_ekg |
| 37 | $D_{4}$ | \( ( 1 + 7 T + 76 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.o_ht_ciw_ryy |
| 41 | $D_{4}$ | \( ( 1 - 10 T + 66 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.au_iy_adei_xog |
| 43 | $C_2^2 \wr C_2$ | \( 1 - 89 T^{2} + 4848 T^{4} - 89 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_adl_a_hem |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 21 T^{2} + 4518 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_v_a_gru |
| 53 | $D_{4}$ | \( ( 1 + 10 T + 90 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.u_ku_ega_bjzi |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 130 T^{2} + 9178 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_fa_a_npa |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) | 4.61.a_jk_a_bhas |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 66 T^{2} + 10026 T^{4} - 66 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_aco_a_ovq |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 237 T^{2} + 23622 T^{4} + 237 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_jd_a_biyo |
| 73 | $D_{4}$ | \( ( 1 - 2 T + 106 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.ae_ii_abbo_bhgo |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 128 T^{2} + 8542 T^{4} + 128 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_ey_a_mqo |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 238 T^{2} + 25930 T^{4} + 238 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_je_a_bmji |
| 89 | $D_{4}$ | \( ( 1 - 10 T + 162 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.au_qi_ahlc_dkpi |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{4} \) | 4.97.abo_bma_axdo_kkmw |
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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
−6.20826212317364981559551151975, −5.81015386992286620815371435990, −5.72078407464534970419689778267, −5.51210847023857145033314373997, −5.43303777992804807394869231460, −5.04184366675578037890601778628, −4.88002403687181117842021479863, −4.86267824858007975268894449355, −4.43266219631839736848393584400, −4.30372538607335411763684824654, −4.15083850789490504304277681718, −3.74461143985009031212979111242, −3.53461795513367244313317904301, −3.20172232636374848922851748263, −3.17836463439786396408546822860, −2.98175719542724164014700988459, −2.97633421420419566238429508069, −2.34833689272783114998014315397, −2.04647040162243789782065354916, −1.84034337203743245278163591478, −1.55421736171288589296071946348, −1.51655396822114015307112104229, −0.977499633894826382732130578315, −0.71478006247534251599436837326, −0.52368106499534165931989268837,
0.52368106499534165931989268837, 0.71478006247534251599436837326, 0.977499633894826382732130578315, 1.51655396822114015307112104229, 1.55421736171288589296071946348, 1.84034337203743245278163591478, 2.04647040162243789782065354916, 2.34833689272783114998014315397, 2.97633421420419566238429508069, 2.98175719542724164014700988459, 3.17836463439786396408546822860, 3.20172232636374848922851748263, 3.53461795513367244313317904301, 3.74461143985009031212979111242, 4.15083850789490504304277681718, 4.30372538607335411763684824654, 4.43266219631839736848393584400, 4.86267824858007975268894449355, 4.88002403687181117842021479863, 5.04184366675578037890601778628, 5.43303777992804807394869231460, 5.51210847023857145033314373997, 5.72078407464534970419689778267, 5.81015386992286620815371435990, 6.20826212317364981559551151975
