| L(s) = 1 | − 2·9-s + 8·17-s + 16·23-s + 8·25-s + 8·41-s − 16·47-s − 24·49-s + 64·79-s + 3·81-s + 24·89-s − 8·97-s − 32·103-s − 8·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·153-s + 157-s + 163-s + 167-s + 28·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2/3·9-s + 1.94·17-s + 3.33·23-s + 8/5·25-s + 1.24·41-s − 2.33·47-s − 3.42·49-s + 7.20·79-s + 1/3·81-s + 2.54·89-s − 0.812·97-s − 3.15·103-s − 0.752·113-s + 1.81·121-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 − 1.29·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.15·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.599117714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.599117714\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 34 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 214 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 406 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 1794 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 12 T^{2} - 1834 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7030 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 104 T^{2} + 5730 T^{4} - 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9814 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 108 T^{2} + 5750 T^{4} - 108 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 32 T + 412 T^{2} - 32 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 244 T^{2} + 27510 T^{4} - 244 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_4$ | \( ( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.63008454217086232538018307980, −6.55419877521952211605275077544, −6.48619350649292541050087208352, −6.12796822915515131172081834959, −5.91652620046539319883213307928, −5.59351933969330532979706683902, −5.20013157075834892367087682760, −5.16767976505098045935310863709, −5.15626871804613970961231341687, −4.79199220276005773722625936669, −4.61474190094363149758358777689, −4.58296345990859042783616077951, −3.84413931797163736311661295286, −3.74527519134794824439130655578, −3.37170069424412586585532852100, −3.30945680020967688693928825892, −3.05894501345611092851825850838, −2.95040778988887081601800938075, −2.61258961360982636176049801494, −2.12126073077874507493882946325, −1.98688316324934969795210808010, −1.38356084350005704101408452395, −1.16834594402284063180567036001, −0.858203707992224204040824148929, −0.48691453854691356156829063829,
0.48691453854691356156829063829, 0.858203707992224204040824148929, 1.16834594402284063180567036001, 1.38356084350005704101408452395, 1.98688316324934969795210808010, 2.12126073077874507493882946325, 2.61258961360982636176049801494, 2.95040778988887081601800938075, 3.05894501345611092851825850838, 3.30945680020967688693928825892, 3.37170069424412586585532852100, 3.74527519134794824439130655578, 3.84413931797163736311661295286, 4.58296345990859042783616077951, 4.61474190094363149758358777689, 4.79199220276005773722625936669, 5.15626871804613970961231341687, 5.16767976505098045935310863709, 5.20013157075834892367087682760, 5.59351933969330532979706683902, 5.91652620046539319883213307928, 6.12796822915515131172081834959, 6.48619350649292541050087208352, 6.55419877521952211605275077544, 6.63008454217086232538018307980
