| L(s) = 1 | + 8·7-s + 8·17-s − 16·23-s + 8·25-s − 24·31-s − 24·41-s + 16·47-s + 16·49-s − 48·71-s − 16·73-s − 40·79-s + 8·89-s + 8·97-s + 8·103-s − 24·113-s + 64·119-s + 36·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 128·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.02·7-s + 1.94·17-s − 3.33·23-s + 8/5·25-s − 4.31·31-s − 3.74·41-s + 2.33·47-s + 16/7·49-s − 5.69·71-s − 1.87·73-s − 4.50·79-s + 0.847·89-s + 0.812·97-s + 0.788·103-s − 2.25·113-s + 5.86·119-s + 3.27·121-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 − 10.0·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.438374479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.438374479\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_4$ | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 4354 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 52 T^{2} + 1366 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_4$ | \( ( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3094 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 4066 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 10870 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 8 T + 130 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 20 T + 240 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 68 T^{2} + 12886 T^{4} - 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 97 | $D_{4}$ | \( ( 1 - 4 T + 70 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−5.75968445501680270211203448855, −5.58396203307732431183139088584, −5.57031318318754740468010663955, −5.56477740766218891017586581507, −5.00271437312430610242654086205, −4.75387054156745692400070625279, −4.72336500395271135453726009694, −4.64417730919133032585895679480, −4.21729156958271808481788530140, −4.14907299330438271386954608392, −4.10679164714186094572481860631, −3.64758591128935322869634687045, −3.30481885441278746030408684627, −3.28572263298868847782614675995, −3.12153925006333445692482880975, −2.90882498536768087829469425631, −2.46868646195459125563240609431, −2.03641760690677580982251372192, −1.80003910916172094604559066927, −1.74728279268091215083279673267, −1.70647320274923551760404857302, −1.38929582007157612389015645166, −1.28978898462738994576129746431, −0.48957208791820352836539566835, −0.27711565706114710016875860092,
0.27711565706114710016875860092, 0.48957208791820352836539566835, 1.28978898462738994576129746431, 1.38929582007157612389015645166, 1.70647320274923551760404857302, 1.74728279268091215083279673267, 1.80003910916172094604559066927, 2.03641760690677580982251372192, 2.46868646195459125563240609431, 2.90882498536768087829469425631, 3.12153925006333445692482880975, 3.28572263298868847782614675995, 3.30481885441278746030408684627, 3.64758591128935322869634687045, 4.10679164714186094572481860631, 4.14907299330438271386954608392, 4.21729156958271808481788530140, 4.64417730919133032585895679480, 4.72336500395271135453726009694, 4.75387054156745692400070625279, 5.00271437312430610242654086205, 5.56477740766218891017586581507, 5.57031318318754740468010663955, 5.58396203307732431183139088584, 5.75968445501680270211203448855
