| L(s) = 1 | − 3-s + 2·5-s + 5·7-s − 9-s + 2·11-s − 6·13-s − 2·15-s − 3·17-s − 7·19-s − 5·21-s − 6·23-s + 3·25-s − 13·29-s − 7·31-s − 2·33-s + 10·35-s − 19·37-s + 6·39-s − 8·41-s − 2·43-s − 2·45-s + 2·47-s + 9·49-s + 3·51-s − 7·53-s + 4·55-s + 7·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.88·7-s − 1/3·9-s + 0.603·11-s − 1.66·13-s − 0.516·15-s − 0.727·17-s − 1.60·19-s − 1.09·21-s − 1.25·23-s + 3/5·25-s − 2.41·29-s − 1.25·31-s − 0.348·33-s + 1.69·35-s − 3.12·37-s + 0.960·39-s − 1.24·41-s − 0.304·43-s − 0.298·45-s + 0.291·47-s + 9/7·49-s + 0.420·51-s − 0.961·53-s + 0.539·55-s + 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 7 T + 46 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7 T + 70 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 19 T + 160 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 70 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T - 58 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 80 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 21 T + 228 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 138 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 7 T + 152 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 18 T + 258 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
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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
−8.323784962239058260932779853742, −8.234443550820800128405297268685, −7.54222154611519566183155398959, −7.20656559809655663097469257924, −6.93276560760917416372114472564, −6.63678085583965776916379132062, −5.88917318315164901140996547734, −5.79679371114517660548170494778, −5.28043030800797253275426450313, −5.01539972206184481208126646083, −4.80672524021410870161768311791, −4.18988952178953854411895160969, −3.77138335136322804844594958431, −3.41871794220961589222237161623, −2.34807364501292709245261122395, −2.05424998490563492749081723396, −1.90836456464008613511330904041, −1.48379682838165193873615172822, 0, 0,
1.48379682838165193873615172822, 1.90836456464008613511330904041, 2.05424998490563492749081723396, 2.34807364501292709245261122395, 3.41871794220961589222237161623, 3.77138335136322804844594958431, 4.18988952178953854411895160969, 4.80672524021410870161768311791, 5.01539972206184481208126646083, 5.28043030800797253275426450313, 5.79679371114517660548170494778, 5.88917318315164901140996547734, 6.63678085583965776916379132062, 6.93276560760917416372114472564, 7.20656559809655663097469257924, 7.54222154611519566183155398959, 8.234443550820800128405297268685, 8.323784962239058260932779853742
