| L(s) = 1 | + 6·4-s − 16·5-s + 15·16-s − 96·20-s + 20·23-s + 60·25-s + 52·31-s − 100·47-s + 96·49-s + 180·53-s − 58·59-s + 60·64-s − 230·67-s − 28·71-s − 240·80-s − 122·89-s + 120·92-s − 10·97-s + 360·100-s + 120·103-s + 110·113-s − 320·115-s + 312·124-s + 720·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 3.19·5-s + 0.937·16-s − 4.79·20-s + 0.869·23-s + 12/5·25-s + 1.67·31-s − 2.12·47-s + 1.95·49-s + 3.39·53-s − 0.983·59-s + 0.937·64-s − 3.43·67-s − 0.394·71-s − 3·80-s − 1.37·89-s + 1.30·92-s − 0.103·97-s + 18/5·100-s + 1.16·103-s + 0.973·113-s − 2.78·115-s + 2.51·124-s + 5.75·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.522704984\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.522704984\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2:C_4$ | \( 1 - 3 p T^{2} + 21 T^{4} - 3 p^{5} T^{6} + p^{8} T^{8} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{4} \) |
| 7 | $C_2^2:C_4$ | \( 1 - 96 T^{2} + 4686 T^{4} - 96 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 576 T^{2} + 139566 T^{4} - 576 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 311 T^{2} + 155521 T^{4} - 311 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 1299 T^{2} + 677841 T^{4} - 1299 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 10 T + 1078 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 2544 T^{2} + 2874126 T^{4} - 2544 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 26 T + 1686 T^{2} - 26 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 1118 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 1799 T^{2} + 241 p^{2} T^{4} - 1799 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $C_2^2:C_4$ | \( 1 - 5771 T^{2} + 15084961 T^{4} - 5771 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 50 T + 5038 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 90 T + 6518 T^{2} - 90 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_{4}$ | \( ( 1 + 29 T + 6891 T^{2} + 29 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 13584 T^{2} + 73803726 T^{4} - 13584 p^{4} T^{6} + p^{8} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 115 T + 11923 T^{2} + 115 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 14 T + 5326 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 1871 T^{2} + 56932441 T^{4} - 1871 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 20604 T^{2} + 179346246 T^{4} - 20604 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 17411 T^{2} + 167253721 T^{4} - 17411 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 61 T + 8161 T^{2} + 61 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 5 T + 13213 T^{2} + 5 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−7.06808442974236558010489455439, −6.73760258851601952265772183979, −6.56901894183215736128440586844, −6.09103962341931874258136036438, −6.06982995329676931090903618175, −5.80510360431227045890444149683, −5.67311910389124046665553135231, −5.21194140649863432990723560215, −4.90423325069693016767500345821, −4.64053413427168541068576518267, −4.59741990788380711584796739462, −4.06411913434107265078866979670, −3.96483852507239753342913204407, −3.85362431010511985549130568436, −3.75334420748338234436831939892, −3.14531416173286865224890021001, −2.99807977384871360239025297522, −2.77965329626048017368805315534, −2.66452834322818774315286715840, −2.04959420830303774468371384474, −1.73182039755478246968776387482, −1.59258987529577549039679450988, −0.863041628387778892829239717971, −0.53237918866483149563176950493, −0.36465097085984279036202975817,
0.36465097085984279036202975817, 0.53237918866483149563176950493, 0.863041628387778892829239717971, 1.59258987529577549039679450988, 1.73182039755478246968776387482, 2.04959420830303774468371384474, 2.66452834322818774315286715840, 2.77965329626048017368805315534, 2.99807977384871360239025297522, 3.14531416173286865224890021001, 3.75334420748338234436831939892, 3.85362431010511985549130568436, 3.96483852507239753342913204407, 4.06411913434107265078866979670, 4.59741990788380711584796739462, 4.64053413427168541068576518267, 4.90423325069693016767500345821, 5.21194140649863432990723560215, 5.67311910389124046665553135231, 5.80510360431227045890444149683, 6.06982995329676931090903618175, 6.09103962341931874258136036438, 6.56901894183215736128440586844, 6.73760258851601952265772183979, 7.06808442974236558010489455439
