| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s − 6·7-s + 4·8-s − 4·10-s + 4·11-s − 4·13-s − 12·14-s + 5·16-s − 4·17-s + 2·19-s − 6·20-s + 8·22-s + 16·23-s + 3·25-s − 8·26-s − 18·28-s + 4·29-s − 2·31-s + 6·32-s − 8·34-s + 12·35-s + 2·37-s + 4·38-s − 8·40-s + 4·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s − 2.26·7-s + 1.41·8-s − 1.26·10-s + 1.20·11-s − 1.10·13-s − 3.20·14-s + 5/4·16-s − 0.970·17-s + 0.458·19-s − 1.34·20-s + 1.70·22-s + 3.33·23-s + 3/5·25-s − 1.56·26-s − 3.40·28-s + 0.742·29-s − 0.359·31-s + 1.06·32-s − 1.37·34-s + 2.02·35-s + 0.328·37-s + 0.648·38-s − 1.26·40-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11088900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.376677806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.376677806\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 26 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 60 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 100 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 10 T + 116 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10 T + 84 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 14 T + 204 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 212 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.988876113344153921594680461266, −8.451375181889865509583324277715, −7.918902895858996860740536914729, −7.36680006553795715284092477221, −7.09758896522786767159942168519, −6.75584917677428637725725806710, −6.61529116519936591247028677725, −6.47428863621165422036753705211, −5.62869059140768206230099576192, −5.42330427898711946486797224180, −4.84498498111578633097509312256, −4.66297590856814571721982725487, −3.88609616880719151677548468068, −3.88201525601083615060169804748, −3.19349904571341647880365677377, −3.18383948621967498861679021003, −2.39797329385675176809751330251, −2.31923939303283907845607874635, −0.859662100440955635595558502502, −0.76184886513914383464049980410,
0.76184886513914383464049980410, 0.859662100440955635595558502502, 2.31923939303283907845607874635, 2.39797329385675176809751330251, 3.18383948621967498861679021003, 3.19349904571341647880365677377, 3.88201525601083615060169804748, 3.88609616880719151677548468068, 4.66297590856814571721982725487, 4.84498498111578633097509312256, 5.42330427898711946486797224180, 5.62869059140768206230099576192, 6.47428863621165422036753705211, 6.61529116519936591247028677725, 6.75584917677428637725725806710, 7.09758896522786767159942168519, 7.36680006553795715284092477221, 7.918902895858996860740536914729, 8.451375181889865509583324277715, 8.988876113344153921594680461266
