| L(s) = 1 | + 2-s − 7·4-s − 9·7-s − 7·8-s − 37·11-s + 112·13-s − 9·14-s − 7·16-s − 77·17-s + 35·19-s − 37·22-s + 267·23-s + 112·26-s + 63·28-s + 325·29-s − 12·31-s − 71·32-s − 77·34-s + 638·37-s + 35·38-s + 238·41-s + 97·43-s + 259·44-s + 267·46-s + 901·47-s − 617·49-s − 784·52-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s − 0.485·7-s − 0.309·8-s − 1.01·11-s + 2.38·13-s − 0.171·14-s − 0.109·16-s − 1.09·17-s + 0.422·19-s − 0.358·22-s + 2.42·23-s + 0.844·26-s + 0.425·28-s + 2.08·29-s − 0.0695·31-s − 0.392·32-s − 0.388·34-s + 2.83·37-s + 0.149·38-s + 0.906·41-s + 0.344·43-s + 0.887·44-s + 0.855·46-s + 2.79·47-s − 1.79·49-s − 2.09·52-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
| \(L(2)\) |
\(\approx\) |
\(4.438765904\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.438765904\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 - T + p^{3} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 9 T + 698 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 37 T + 2798 T^{2} + 37 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 112 T + 7002 T^{2} - 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 77 T + 6152 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 35 T + 8868 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 267 T + 37792 T^{2} - 267 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 325 T + 62636 T^{2} - 325 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p^{3} T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 638 T + 5466 p T^{2} - 638 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 238 T + 2983 p T^{2} - 238 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 97 T + 98922 T^{2} - 97 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 901 T + 409598 T^{2} - 901 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 224 T + 227798 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 85 T + 392756 T^{2} + 85 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 247 T + 155508 T^{2} + 247 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 606 T + 414023 T^{2} - 606 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 394 T + 654806 T^{2} - 394 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 811 T + 595758 T^{2} - 811 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 840 T + 1114826 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 387 T - 511958 T^{2} - 387 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1065 T + 874888 T^{2} + 1065 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1031 T + 1949508 T^{2} - 1031 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.008073275716396023336573338230, −8.721430606223867360807509826450, −8.082991272468497771232226930528, −8.077003872408058581943931560220, −7.48874712580067686935627701639, −6.83812209575370133284435636229, −6.54233293593491325873856451036, −6.31727547066375693355101709157, −5.65950633301496874847148803760, −5.48259997648817616059998114805, −4.76080242905033430059135946428, −4.60273768479772032609598759580, −4.11238155225051398893582542526, −3.75559408520601284265785133118, −2.94806444103018037542449608547, −2.91354525575645104440175042707, −2.31432648459520457654047723882, −1.32726977450677331854385334794, −0.75953583764221999012311352427, −0.63371836511946848181121265426,
0.63371836511946848181121265426, 0.75953583764221999012311352427, 1.32726977450677331854385334794, 2.31432648459520457654047723882, 2.91354525575645104440175042707, 2.94806444103018037542449608547, 3.75559408520601284265785133118, 4.11238155225051398893582542526, 4.60273768479772032609598759580, 4.76080242905033430059135946428, 5.48259997648817616059998114805, 5.65950633301496874847148803760, 6.31727547066375693355101709157, 6.54233293593491325873856451036, 6.83812209575370133284435636229, 7.48874712580067686935627701639, 8.077003872408058581943931560220, 8.082991272468497771232226930528, 8.721430606223867360807509826450, 9.008073275716396023336573338230
