| L(s) = 1 | + 4·5-s + 4·13-s + 24·17-s + 10·25-s − 16·37-s + 8·41-s − 4·49-s − 8·53-s − 32·61-s + 16·65-s + 16·73-s + 10·81-s + 96·85-s + 24·89-s − 8·97-s − 24·101-s − 8·109-s + 8·113-s − 24·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.10·13-s + 5.82·17-s + 2·25-s − 2.63·37-s + 1.24·41-s − 4/7·49-s − 1.09·53-s − 4.09·61-s + 1.98·65-s + 1.87·73-s + 10/9·81-s + 10.4·85-s + 2.54·89-s − 0.812·97-s − 2.38·101-s − 0.766·109-s + 0.752·113-s − 2.18·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.77752243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.77752243\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) | |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) | |
| good | 3 | $D_4\times C_2$ | \( 1 - 10 T^{4} + p^{4} T^{8} \) | 4.3.a_a_a_ak |
| 7 | $D_4\times C_2$ | \( 1 + 4 T^{2} - 10 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} \) | 4.7.a_e_a_ak |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 24 T^{2} + 358 T^{4} + 24 p^{2} T^{6} + p^{4} T^{8} \) | 4.11.a_y_a_nu |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) | 4.17.ay_ky_adci_pja |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 56 T^{2} + 1478 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) | 4.19.a_ce_a_cew |
| 23 | $C_2^2 \wr C_2$ | \( 1 - 16 T^{2} + 1094 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) | 4.23.a_aq_a_bqc |
| 29 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) | 4.29.a_ci_a_dvi |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 72 T^{2} + 3190 T^{4} + 72 p^{2} T^{6} + p^{4} T^{8} \) | 4.31.a_cu_a_ess |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.37.q_hg_ciy_qti |
| 41 | $D_{4}$ | \( ( 1 - 4 T - 26 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.41.ai_abk_aeq_hxu |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 64 T^{2} + 2454 T^{4} + 64 p^{2} T^{6} + p^{4} T^{8} \) | 4.43.a_cm_a_dqk |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 164 T^{2} + 11030 T^{4} + 164 p^{2} T^{6} + p^{4} T^{8} \) | 4.47.a_gi_a_qig |
| 53 | $D_{4}$ | \( ( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.53.i_m_ps_kvm |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 56 T^{2} + 6374 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) | 4.59.a_ce_a_jle |
| 61 | $D_{4}$ | \( ( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.bg_wa_kjo_dqdq |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 228 T^{2} + 21862 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} \) | 4.67.a_iu_a_bgiw |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 104 T^{2} + 10518 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) | 4.71.a_ea_a_poo |
| 73 | $D_{4}$ | \( ( 1 - 8 T + 134 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) | 4.73.aq_mu_aexk_cedy |
| 79 | $C_2^2 \wr C_2$ | \( 1 + 236 T^{2} + 25958 T^{4} + 236 p^{2} T^{6} + p^{4} T^{8} \) | 4.79.a_jc_a_bmkk |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 308 T^{2} + 37382 T^{4} + 308 p^{2} T^{6} + p^{4} T^{8} \) | 4.83.a_lw_a_cdhu |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) | 4.89.ay_nk_agui_cyti |
| 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
−6.03487488681387421024818226974, −5.67695696738140108952765391131, −5.45139703480143022345338430303, −5.39829013796956619678085570557, −5.39158544571761660569391187527, −5.00046772720173259599794708494, −4.86654012853492949299484981974, −4.80278733411239058468275421810, −4.36294999114558105211173905018, −4.03315400870116633074188006048, −3.70531691285854824896158486938, −3.68773477119110878756689717884, −3.54722870052726693278251884042, −3.07141446738064983455625437066, −3.05258154946864322652578549970, −3.00487222002321927227799271693, −2.82809358308422103685722007854, −2.14693831598345330449590498533, −2.08008735585280501823532460786, −1.73890657273346339745969879906, −1.41366873843084005923733511954, −1.29601947823929830137500393836, −1.23059180933521235920959965148, −0.811652905642349258769064079498, −0.41597069085635448261676215685,
0.41597069085635448261676215685, 0.811652905642349258769064079498, 1.23059180933521235920959965148, 1.29601947823929830137500393836, 1.41366873843084005923733511954, 1.73890657273346339745969879906, 2.08008735585280501823532460786, 2.14693831598345330449590498533, 2.82809358308422103685722007854, 3.00487222002321927227799271693, 3.05258154946864322652578549970, 3.07141446738064983455625437066, 3.54722870052726693278251884042, 3.68773477119110878756689717884, 3.70531691285854824896158486938, 4.03315400870116633074188006048, 4.36294999114558105211173905018, 4.80278733411239058468275421810, 4.86654012853492949299484981974, 5.00046772720173259599794708494, 5.39158544571761660569391187527, 5.39829013796956619678085570557, 5.45139703480143022345338430303, 5.67695696738140108952765391131, 6.03487488681387421024818226974
