| L(s) = 1 | + 2·5-s − 4·7-s − 3·9-s − 4·19-s − 8·23-s + 3·25-s + 4·29-s − 4·31-s − 8·35-s − 8·37-s + 10·41-s − 6·45-s + 4·47-s + 49-s − 8·53-s − 4·59-s + 10·61-s + 12·63-s − 8·67-s − 12·71-s + 16·73-s − 8·79-s + 8·83-s + 18·89-s − 8·95-s − 8·97-s + 2·101-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.51·7-s − 9-s − 0.917·19-s − 1.66·23-s + 3/5·25-s + 0.742·29-s − 0.718·31-s − 1.35·35-s − 1.31·37-s + 1.56·41-s − 0.894·45-s + 0.583·47-s + 1/7·49-s − 1.09·53-s − 0.520·59-s + 1.28·61-s + 1.51·63-s − 0.977·67-s − 1.42·71-s + 1.87·73-s − 0.900·79-s + 0.878·83-s + 1.90·89-s − 0.820·95-s − 0.812·97-s + 0.199·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93702400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 15 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 + 8 T + 50 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 83 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 110 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 10 T + 135 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 147 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 170 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 211 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−7.54958666253750321400641172001, −7.04102364159000534145249811549, −6.58001133805252223183753547120, −6.54701490297771638631014646471, −6.08505146326523196633580343295, −5.98237436244155856162371504214, −5.46270060013173614124114093688, −5.36761139391715308369769308583, −4.70880877042651724065276724896, −4.39711746878523054302882014373, −3.88313811751649704228374111492, −3.62389442896025403938146172801, −3.10888668579405289596584549407, −2.88244464867855645189950268341, −2.34249944122802168856153007463, −2.11140615330849230066908900021, −1.58420710595563443477132095832, −0.895319479502240954358746804788, 0, 0,
0.895319479502240954358746804788, 1.58420710595563443477132095832, 2.11140615330849230066908900021, 2.34249944122802168856153007463, 2.88244464867855645189950268341, 3.10888668579405289596584549407, 3.62389442896025403938146172801, 3.88313811751649704228374111492, 4.39711746878523054302882014373, 4.70880877042651724065276724896, 5.36761139391715308369769308583, 5.46270060013173614124114093688, 5.98237436244155856162371504214, 6.08505146326523196633580343295, 6.54701490297771638631014646471, 6.58001133805252223183753547120, 7.04102364159000534145249811549, 7.54958666253750321400641172001
