| L(s) = 1 | − 6·3-s − 24·5-s + 27·9-s − 44·11-s + 26·13-s + 144·15-s + 164·17-s + 48·19-s − 8·23-s + 238·25-s − 108·27-s − 404·29-s − 40·31-s + 264·33-s + 100·37-s − 156·39-s + 200·41-s − 616·43-s − 648·45-s + 324·47-s − 630·49-s − 984·51-s + 164·53-s + 1.05e3·55-s − 288·57-s + 140·59-s − 628·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.14·5-s + 9-s − 1.20·11-s + 0.554·13-s + 2.47·15-s + 2.33·17-s + 0.579·19-s − 0.0725·23-s + 1.90·25-s − 0.769·27-s − 2.58·29-s − 0.231·31-s + 1.39·33-s + 0.444·37-s − 0.640·39-s + 0.761·41-s − 2.18·43-s − 2.14·45-s + 1.00·47-s − 1.83·49-s − 2.70·51-s + 0.425·53-s + 2.58·55-s − 0.669·57-s + 0.308·59-s − 1.31·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6230016 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8031320730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8031320730\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 13 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 24 T + 338 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 90 p T^{2} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 p T + 1130 T^{2} + 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 164 T + 16326 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 48 T + 14238 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 T - 7906 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 404 T + 81518 T^{2} + 404 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 40 T + 50518 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 100 T + 92830 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 200 T + 24138 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 616 T + 216022 T^{2} + 616 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 324 T + 219554 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 164 T + 102878 T^{2} - 164 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 140 T + 393258 T^{2} - 140 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 628 T + 293614 T^{2} + 628 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 472 T + 252622 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 428 T + 662834 T^{2} + 428 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 900 T + 899670 T^{2} + 900 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 432 T + 924318 T^{2} - 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1388 T + 1567866 T^{2} + 1388 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 960 T + 1134938 T^{2} - 960 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 532 T + 1218502 T^{2} + 532 p^{3} T^{3} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.545361469898863769993654433358, −8.214749650847019865179813266684, −7.83311175650407435717480886285, −7.58286149622746433552806801717, −7.29609653330291740539923491394, −7.21092013855399311097742142626, −6.36556308297242734530668123142, −5.87304165299979100831131316746, −5.65230746249493606781413848497, −5.35715813829180051299159178833, −4.68904464806183213011307753558, −4.58063077609859076340659378902, −3.81951254332740743580081873021, −3.59231367417770036023296921716, −3.27241873888482523873536041462, −2.79407484006407047849924043753, −1.69210227831788071392018475629, −1.44942941970077536897394547861, −0.44383129657166711667091732709, −0.44045386976237497450364769373,
0.44045386976237497450364769373, 0.44383129657166711667091732709, 1.44942941970077536897394547861, 1.69210227831788071392018475629, 2.79407484006407047849924043753, 3.27241873888482523873536041462, 3.59231367417770036023296921716, 3.81951254332740743580081873021, 4.58063077609859076340659378902, 4.68904464806183213011307753558, 5.35715813829180051299159178833, 5.65230746249493606781413848497, 5.87304165299979100831131316746, 6.36556308297242734530668123142, 7.21092013855399311097742142626, 7.29609653330291740539923491394, 7.58286149622746433552806801717, 7.83311175650407435717480886285, 8.214749650847019865179813266684, 8.545361469898863769993654433358
