| L(s) = 1 | + 20·5-s − 6·9-s − 32·13-s − 20·17-s + 192·25-s − 56·29-s − 208·37-s − 92·41-s − 120·45-s + 112·53-s − 168·61-s − 640·65-s + 32·73-s + 27·81-s − 400·85-s − 236·89-s + 304·97-s − 164·101-s + 488·109-s + 136·113-s + 192·117-s + 80·121-s + 1.26e3·125-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 4·5-s − 2/3·9-s − 2.46·13-s − 1.17·17-s + 7.67·25-s − 1.93·29-s − 5.62·37-s − 2.24·41-s − 8/3·45-s + 2.11·53-s − 2.75·61-s − 9.84·65-s + 0.438·73-s + 1/3·81-s − 4.70·85-s − 2.65·89-s + 3.13·97-s − 1.62·101-s + 4.47·109-s + 1.20·113-s + 1.64·117-s + 0.661·121-s + 10.0·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.242560690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.242560690\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 2 p T + 54 T^{2} - 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 11982 T^{4} - 80 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 16 T + 318 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 10 T + 582 T^{2} + 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 274 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 65742 T^{4} - 128 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 28 T + 1542 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 1490 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 104 T + 5358 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 46 T + 2862 T^{2} + 46 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6116 T^{2} + 15844902 T^{4} - 6116 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 8308 T^{2} + 27002982 T^{4} - 8308 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 56 T + 2286 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 1940 T^{2} + 10358022 T^{4} - 1940 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 84 T + 7862 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 + 44 T^{2} + 31484742 T^{4} + 44 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 13280 T^{2} + 92626062 T^{4} - 13280 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 16 T + 8622 T^{2} - 16 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 17524 T^{2} + 151177062 T^{4} - 17524 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 13124 T^{2} + 100043430 T^{4} - 13124 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 118 T + 18798 T^{2} + 118 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 - 152 T + 22494 T^{2} - 152 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.23406986879705690516113504977, −5.82282639174542888148321607909, −5.80114386376110006113944256742, −5.45739372513641759987434523339, −5.37841282198820942285490473684, −5.29700892015897448170576304935, −5.16750873661608667414365502221, −4.77721774664838390835077003552, −4.65680888267060717074813351598, −4.56134398525228240916183845795, −4.13559220699278787562756502109, −3.52987317870867121183152822677, −3.52548102150223635396273207390, −3.27712210845326381328339041855, −3.22529575488887535204700429221, −2.57207772985668609739763713094, −2.31751462900379023057927650987, −2.30683550519769809177735835606, −2.22959992671823236685329313818, −1.79689007383400274171807634961, −1.65205393825054313573273720860, −1.58407128472493898709315412890, −1.17436538884681563788121685552, −0.32416634959817957926867368140, −0.16982063156988279293594728730,
0.16982063156988279293594728730, 0.32416634959817957926867368140, 1.17436538884681563788121685552, 1.58407128472493898709315412890, 1.65205393825054313573273720860, 1.79689007383400274171807634961, 2.22959992671823236685329313818, 2.30683550519769809177735835606, 2.31751462900379023057927650987, 2.57207772985668609739763713094, 3.22529575488887535204700429221, 3.27712210845326381328339041855, 3.52548102150223635396273207390, 3.52987317870867121183152822677, 4.13559220699278787562756502109, 4.56134398525228240916183845795, 4.65680888267060717074813351598, 4.77721774664838390835077003552, 5.16750873661608667414365502221, 5.29700892015897448170576304935, 5.37841282198820942285490473684, 5.45739372513641759987434523339, 5.80114386376110006113944256742, 5.82282639174542888148321607909, 6.23406986879705690516113504977
