| L(s) = 1 | − 3·4-s + 9-s + 4·11-s + 4·16-s − 12·19-s − 2·29-s + 16·31-s − 3·36-s − 18·41-s − 12·44-s − 10·49-s − 2·61-s − 9·64-s − 12·71-s + 36·76-s + 16·79-s − 28·89-s + 4·99-s − 6·101-s + 8·109-s + 6·116-s + 26·121-s − 48·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 3/2·4-s + 1/3·9-s + 1.20·11-s + 16-s − 2.75·19-s − 0.371·29-s + 2.87·31-s − 1/2·36-s − 2.81·41-s − 1.80·44-s − 1.42·49-s − 0.256·61-s − 9/8·64-s − 1.42·71-s + 4.12·76-s + 1.80·79-s − 2.96·89-s + 0.402·99-s − 0.597·101-s + 0.766·109-s + 0.557·116-s + 2.36·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1291879380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1291879380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 15 T^{2} - 64 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2^2$ | \( ( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 50 T^{2} + 651 T^{4} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 130 T^{2} + 12411 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−7.25019057858392844697250338907, −6.69668876118605857753454877760, −6.64507643093177295967515543166, −6.53939587162967681415380883043, −6.37731148404106877111724777158, −6.18234102616990394911301502534, −5.75940680302604132082835187481, −5.48119836623663316267850171462, −5.43976872868083027408557944012, −4.82513111239089010478084671923, −4.76767169821405570355701048215, −4.52237486564657571695483921038, −4.39558177236465630605022397443, −4.34886751371994639087168668168, −3.80714338780038580501691709636, −3.61760704821086556426416232742, −3.52965595803869456353753882088, −2.91088989574906204107096125365, −2.89233788946352535354654974379, −2.34985677678625514229708911618, −2.01884020059328445376895218204, −1.60902802844412754985893236847, −1.38991240527399650658459833074, −0.880111799695273261095212930533, −0.10036757115200319802414967059,
0.10036757115200319802414967059, 0.880111799695273261095212930533, 1.38991240527399650658459833074, 1.60902802844412754985893236847, 2.01884020059328445376895218204, 2.34985677678625514229708911618, 2.89233788946352535354654974379, 2.91088989574906204107096125365, 3.52965595803869456353753882088, 3.61760704821086556426416232742, 3.80714338780038580501691709636, 4.34886751371994639087168668168, 4.39558177236465630605022397443, 4.52237486564657571695483921038, 4.76767169821405570355701048215, 4.82513111239089010478084671923, 5.43976872868083027408557944012, 5.48119836623663316267850171462, 5.75940680302604132082835187481, 6.18234102616990394911301502534, 6.37731148404106877111724777158, 6.53939587162967681415380883043, 6.64507643093177295967515543166, 6.69668876118605857753454877760, 7.25019057858392844697250338907
