| L(s) = 1 | − 80·3-s − 1.25e4·9-s + 3.20e4·11-s + 2.48e5·17-s − 2.41e5·19-s + 2.01e4·25-s + 1.29e6·27-s − 2.56e6·33-s + 6.96e6·41-s − 2.26e7·43-s + 5.59e6·49-s − 1.98e7·51-s + 1.93e7·57-s + 2.66e7·59-s − 3.13e7·67-s − 2.97e7·73-s − 1.61e6·75-s + 4.83e7·81-s − 1.39e8·83-s + 4.72e7·89-s − 3.95e8·97-s − 4.01e8·99-s + 4.48e7·107-s − 2.86e8·113-s − 1.41e8·121-s − 5.57e8·123-s + 127-s + ⋯ |
| L(s) = 1 | − 0.987·3-s − 1.90·9-s + 2.18·11-s + 2.97·17-s − 1.85·19-s + 0.0515·25-s + 2.44·27-s − 2.16·33-s + 2.46·41-s − 6.61·43-s + 0.969·49-s − 2.93·51-s + 1.83·57-s + 2.20·59-s − 1.55·67-s − 1.04·73-s − 0.0509·75-s + 1.12·81-s − 2.93·83-s + 0.753·89-s − 4.47·97-s − 4.17·99-s + 0.341·107-s − 1.75·113-s − 0.661·121-s − 2.43·123-s + 6.53·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.06272502303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06272502303\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 + 40 T + 962 p^{2} T^{2} + 40 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 20156 T^{2} + 4281712374 p^{2} T^{4} - 20156 p^{16} T^{6} + p^{32} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 114116 p^{2} T^{2} + 28741685190 p^{4} T^{4} - 114116 p^{18} T^{6} + p^{32} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 16024 T + 456094290 T^{2} - 16024 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 1954707644 T^{2} + 2032392155908111110 T^{4} - 1954707644 p^{16} T^{6} + p^{32} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 124260 T + 12631375046 T^{2} - 124260 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 6360 p T + 1547904422 p T^{2} + 6360 p^{9} T^{3} + p^{16} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 22564164740 T^{2} + \)\(26\!\cdots\!58\)\( T^{4} - 22564164740 p^{16} T^{6} + p^{32} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 315706203068 T^{2} + \)\(47\!\cdots\!54\)\( T^{4} - 315706203068 p^{16} T^{6} + p^{32} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2308308775940 T^{2} + \)\(28\!\cdots\!38\)\( p^{2} T^{4} - 2308308775940 p^{16} T^{6} + p^{32} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 6170532512188 T^{2} + \)\(25\!\cdots\!14\)\( T^{4} - 6170532512188 p^{16} T^{6} + p^{32} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 3482044 T + 18665604055302 T^{2} - 3482044 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 263032 p T + 55349192011602 T^{2} + 263032 p^{9} T^{3} + p^{16} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 19635696904196 T^{2} + \)\(12\!\cdots\!46\)\( T^{4} - 19635696904196 p^{16} T^{6} + p^{32} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 235594728119740 T^{2} + \)\(21\!\cdots\!38\)\( T^{4} - 235594728119740 p^{16} T^{6} + p^{32} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 13340728 T + 320233862225874 T^{2} - 13340728 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 550183808721596 T^{2} + \)\(14\!\cdots\!26\)\( T^{4} - 550183808721596 p^{16} T^{6} + p^{32} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 15695912 T - 138380377387566 T^{2} + 15695912 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 2546570207785604 T^{2} + \)\(24\!\cdots\!10\)\( T^{4} - 2546570207785604 p^{16} T^{6} + p^{32} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 14888676 T + 1528099309486406 T^{2} + 14888676 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 3114613788648452 T^{2} + \)\(67\!\cdots\!18\)\( T^{4} - 3114613788648452 p^{16} T^{6} + p^{32} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 69603080 T + 4484679918867282 T^{2} + 69603080 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 23623452 T + 7124716017813062 T^{2} - 23623452 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 197989660 T + 24238250475556038 T^{2} + 197989660 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
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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
−7.03022936943092412852646518509, −6.94970972900526425822534129745, −6.65216140939551770226223424516, −6.46939798811951662886430935940, −6.23948837390566489386730995774, −5.86261587538012636097570565853, −5.77549469728224534547801611437, −5.37712833649921002207461848378, −5.33756208114487397510709846451, −5.03601728713919266746595768986, −4.66615276168468285148375738403, −4.16071862575880596866572673176, −3.87093723816232967540111218918, −3.85971482415596432710637791840, −3.46452122951130465047916793404, −3.02933568941822678541843461977, −2.75110385836405893625049387256, −2.72405015718139988212913183155, −2.10901194510005523842181921297, −1.64032714423424140353929381125, −1.26173952264267493975581824609, −1.23297732161143763509169761386, −1.03126039810996082775464931636, −0.22496994106356000326632350025, −0.06509752122629093222742040582,
0.06509752122629093222742040582, 0.22496994106356000326632350025, 1.03126039810996082775464931636, 1.23297732161143763509169761386, 1.26173952264267493975581824609, 1.64032714423424140353929381125, 2.10901194510005523842181921297, 2.72405015718139988212913183155, 2.75110385836405893625049387256, 3.02933568941822678541843461977, 3.46452122951130465047916793404, 3.85971482415596432710637791840, 3.87093723816232967540111218918, 4.16071862575880596866572673176, 4.66615276168468285148375738403, 5.03601728713919266746595768986, 5.33756208114487397510709846451, 5.37712833649921002207461848378, 5.77549469728224534547801611437, 5.86261587538012636097570565853, 6.23948837390566489386730995774, 6.46939798811951662886430935940, 6.65216140939551770226223424516, 6.94970972900526425822534129745, 7.03022936943092412852646518509
