| L(s) = 1 | − 2·3-s − 2·4-s + 4·9-s − 6·11-s + 4·12-s + 4·13-s − 5·16-s − 24·17-s − 4·19-s + 12·23-s + 7·25-s − 4·27-s + 6·29-s + 2·31-s + 12·33-s − 8·36-s − 28·37-s − 8·39-s − 10·43-s + 12·44-s − 12·47-s + 10·48-s + 48·51-s − 8·52-s + 6·53-s + 8·57-s + 36·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4-s + 4/3·9-s − 1.80·11-s + 1.15·12-s + 1.10·13-s − 5/4·16-s − 5.82·17-s − 0.917·19-s + 2.50·23-s + 7/5·25-s − 0.769·27-s + 1.11·29-s + 0.359·31-s + 2.08·33-s − 4/3·36-s − 4.60·37-s − 1.28·39-s − 1.52·43-s + 1.80·44-s − 1.75·47-s + 1.44·48-s + 6.72·51-s − 1.10·52-s + 0.824·53-s + 1.05·57-s + 4.68·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4718339118\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4718339118\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$ | \( ( 1 + T^{2} + p^{2} T^{4} )^{2} \) |
| 3 | $D_4\times C_2$ | \( 1 + 2 T - 4 T^{3} - 5 T^{4} - 4 p T^{5} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^3$ | \( 1 - 7 T^{2} + 24 T^{4} - 7 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} + 36 T^{3} + 267 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 12 T + 67 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 6 T + 52 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 T - 32 T^{2} + 52 T^{3} + 211 T^{4} + 52 p T^{5} - 32 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 41 | $C_2^3$ | \( 1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 16 T^{2} - 20 T^{3} + 907 T^{4} - 20 p T^{5} + 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 62 T^{2} - 144 T^{3} - 2253 T^{4} - 144 p T^{5} + 62 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 31 T^{2} + 234 T^{3} - 228 T^{4} + 234 p T^{5} - 31 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 18 T + 196 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 20 T + 205 T^{2} - 1460 T^{3} + 9904 T^{4} - 1460 p T^{5} + 205 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 104 T^{2} + 52 T^{3} + 6907 T^{4} + 52 p T^{5} - 104 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4 T - 107 T^{2} - 92 T^{3} + 8632 T^{4} - 92 p T^{5} - 107 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 22 T + 232 T^{2} + 2068 T^{3} + 19027 T^{4} + 2068 p T^{5} + 232 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 148 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 38 T^{2} + 736 T^{3} - 5213 T^{4} + 736 p T^{5} - 38 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.17089685634967702469706586673, −7.01250330510461880530869810333, −6.98081776627880857927565488118, −6.97875970364840196587077416058, −6.97786485916507294321313482141, −6.45743192805692599652003833837, −6.26678340691436440738378437100, −6.16998439641410739154215555548, −5.53203478773106523627488391067, −5.26901309716399218368244729722, −4.98976548632169577198657356221, −4.86884107384667570159790280736, −4.74434086558219761284269671910, −4.58579823577039771669698339455, −4.49621613864440563568465439889, −3.85535833275129304634712158863, −3.61821477540191044518639051487, −3.51121697870780206351873417530, −2.82887423295651320510993894621, −2.34861426781548045296619249528, −2.31086174803166244226297971791, −2.06608907513741665208850473742, −1.52722091256529924963545745015, −0.49794417300932128086995355946, −0.43518496439568744079557733255,
0.43518496439568744079557733255, 0.49794417300932128086995355946, 1.52722091256529924963545745015, 2.06608907513741665208850473742, 2.31086174803166244226297971791, 2.34861426781548045296619249528, 2.82887423295651320510993894621, 3.51121697870780206351873417530, 3.61821477540191044518639051487, 3.85535833275129304634712158863, 4.49621613864440563568465439889, 4.58579823577039771669698339455, 4.74434086558219761284269671910, 4.86884107384667570159790280736, 4.98976548632169577198657356221, 5.26901309716399218368244729722, 5.53203478773106523627488391067, 6.16998439641410739154215555548, 6.26678340691436440738378437100, 6.45743192805692599652003833837, 6.97786485916507294321313482141, 6.97875970364840196587077416058, 6.98081776627880857927565488118, 7.01250330510461880530869810333, 7.17089685634967702469706586673
