| L(s) = 1 | + 3·2-s − 4-s − 7·7-s − 9·8-s − 66·11-s + 11·13-s − 21·14-s − 9·16-s + 99·17-s − 77·19-s − 198·22-s + 33·23-s + 33·26-s + 7·28-s + 51·29-s + 43·31-s − 153·32-s + 297·34-s + 50·37-s − 231·38-s − 132·41-s − 88·43-s + 66·44-s + 99·46-s + 399·47-s − 575·49-s − 11·52-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 1/8·4-s − 0.377·7-s − 0.397·8-s − 1.80·11-s + 0.234·13-s − 0.400·14-s − 0.140·16-s + 1.41·17-s − 0.929·19-s − 1.91·22-s + 0.299·23-s + 0.248·26-s + 0.0472·28-s + 0.326·29-s + 0.249·31-s − 0.845·32-s + 1.49·34-s + 0.222·37-s − 0.986·38-s − 0.502·41-s − 0.312·43-s + 0.226·44-s + 0.317·46-s + 1.23·47-s − 1.67·49-s − 0.0293·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 3 T + 5 p T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + p T + 624 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 6 p T + 293 p T^{2} + 6 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 11 T + 2568 T^{2} - 11 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 99 T + 11608 T^{2} - 99 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 77 T + 9186 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 33 T + 21628 T^{2} - 33 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 51 T + 49420 T^{2} - 51 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 43 T + 59970 T^{2} - 43 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 50 T + 77874 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 132 T + 62965 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 88 T + 160653 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 399 T + 179128 T^{2} - 399 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 54 T + 81970 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 798 T + 417367 T^{2} + 798 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 439 T + 411186 T^{2} - 439 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 988 T + 809625 T^{2} + 988 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1368 T + 1012606 T^{2} + 1368 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 455 T + 342636 T^{2} - 455 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 803 T + 359562 T^{2} + 803 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 813 T + 1214362 T^{2} - 813 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 396 T + 1406374 T^{2} - 396 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 736 T + 1874937 T^{2} + 736 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.360307700159525156949225280743, −8.329473670039198788071047423341, −7.75254506046487691132432071673, −7.33752666147329640181808497179, −7.20200057680276313778535889193, −6.37312807530202333653737623495, −6.02979336777099538365126385811, −5.83734158005811905783195171427, −5.13820275269862156936450773136, −5.05221691894350691474010169319, −4.55762776552464795565990851977, −4.23388313276644903180198688664, −3.51220965268993638918695374399, −3.29056720081578573632076604100, −2.81394586325117450709485129656, −2.32034612835706651814651496064, −1.62187259801120991777345391730, −0.999741528731994651366378859782, 0, 0,
0.999741528731994651366378859782, 1.62187259801120991777345391730, 2.32034612835706651814651496064, 2.81394586325117450709485129656, 3.29056720081578573632076604100, 3.51220965268993638918695374399, 4.23388313276644903180198688664, 4.55762776552464795565990851977, 5.05221691894350691474010169319, 5.13820275269862156936450773136, 5.83734158005811905783195171427, 6.02979336777099538365126385811, 6.37312807530202333653737623495, 7.20200057680276313778535889193, 7.33752666147329640181808497179, 7.75254506046487691132432071673, 8.329473670039198788071047423341, 8.360307700159525156949225280743
