| L(s) = 1 | − 4·7-s + 8·13-s + 20·37-s + 12·43-s + 10·49-s − 8·61-s + 20·67-s − 32·91-s − 24·97-s − 32·103-s − 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 32·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 2.21·13-s + 3.28·37-s + 1.82·43-s + 10/7·49-s − 1.02·61-s + 2.44·67-s − 3.35·91-s − 2.43·97-s − 3.15·103-s − 0.383·109-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 + 2.46·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.787514263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.787514263\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) | |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) | |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )( 1 + 8 T^{2} + p^{2} T^{4} ) \) | 4.5.a_a_a_ao |
| 11 | $C_2^3$ | \( 1 + 82 T^{4} + p^{4} T^{8} \) | 4.11.a_a_a_de |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.13.ai_bg_agm_bfm |
| 17 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) | 4.17.a_acq_a_cos |
| 19 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.19.a_a_a_bbu |
| 23 | $C_2^2$ | \( ( 1 + 4 T^{2} + p^{2} T^{4} )^{2} \) | 4.23.a_i_a_bpi |
| 29 | $C_2^3$ | \( 1 + 1234 T^{4} + p^{4} T^{8} \) | 4.29.a_a_a_bvm |
| 31 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) | 4.31.a_e_a_cwc |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) | 4.37.au_hs_acoy_ssc |
| 41 | $C_2^2$ | \( ( 1 + 74 T^{2} + p^{2} T^{4} )^{2} \) | 4.41.a_fs_a_nby |
| 43 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) | 4.43.am_cu_abce_knu |
| 47 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) | 4.47.a_eu_a_mfu |
| 53 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) | 4.53.a_a_a_iic |
| 59 | $C_2^3$ | \( 1 + 3442 T^{4} + p^{4} T^{8} \) | 4.59.a_a_a_fck |
| 61 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) | 4.61.i_bg_vg_nzu |
| 67 | $C_2^2$ | \( ( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) | 4.67.au_hs_adma_bkuw |
| 71 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) | 4.71.a_adk_a_rug |
| 73 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.73.a_dw_a_tmc |
| 79 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) | 4.79.a_ho_a_bgrm |
| 83 | $C_2^3$ | \( 1 + 8722 T^{4} + p^{4} T^{8} \) | 4.83.a_a_a_mxm |
| 89 | $C_2^2$ | \( ( 1 + 50 T^{2} + p^{2} T^{4} )^{2} \) | 4.89.a_dw_a_bbdm |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) | 4.97.y_xg_lpw_frkw |
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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
−5.97104767417856032590609226922, −5.78683107609330705098009086812, −5.78653141814735386866876765381, −5.55317703855184712054480741465, −5.16441451975991953119885012224, −4.95901799655736879985017601878, −4.81490029104547902854045541120, −4.48445036447802341088760723222, −4.27748904772342348833274011733, −3.98700823947144538083780697476, −3.85140237752106299450124564437, −3.83011913879812939316983329094, −3.75194804215127376903277288291, −3.19420525361353250450356385950, −3.04876299089640668799702552174, −2.78430318580766510951362466495, −2.63940476403183025939586073807, −2.61876539329111922241849947472, −2.11357097909478692087374615523, −1.83762110196233550844068526979, −1.53546023301149047920337229545, −1.07299328130392228185385731261, −1.01129042407561704034113564159, −0.74321592848274360584647429782, −0.25493654303091938193168758860,
0.25493654303091938193168758860, 0.74321592848274360584647429782, 1.01129042407561704034113564159, 1.07299328130392228185385731261, 1.53546023301149047920337229545, 1.83762110196233550844068526979, 2.11357097909478692087374615523, 2.61876539329111922241849947472, 2.63940476403183025939586073807, 2.78430318580766510951362466495, 3.04876299089640668799702552174, 3.19420525361353250450356385950, 3.75194804215127376903277288291, 3.83011913879812939316983329094, 3.85140237752106299450124564437, 3.98700823947144538083780697476, 4.27748904772342348833274011733, 4.48445036447802341088760723222, 4.81490029104547902854045541120, 4.95901799655736879985017601878, 5.16441451975991953119885012224, 5.55317703855184712054480741465, 5.78653141814735386866876765381, 5.78683107609330705098009086812, 5.97104767417856032590609226922
