| L(s) = 1 | − 3-s + 7-s − 9-s − 2·11-s − 2·13-s + 17-s + 2·19-s − 21-s + 16·23-s − 4·29-s + 8·31-s + 2·33-s − 7·37-s + 2·39-s + 2·41-s − 15·43-s + 13·47-s − 9·49-s − 51-s + 2·53-s − 2·57-s + 2·59-s + 14·61-s − 63-s + 2·67-s − 16·69-s − 3·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.242·17-s + 0.458·19-s − 0.218·21-s + 3.33·23-s − 0.742·29-s + 1.43·31-s + 0.348·33-s − 1.15·37-s + 0.320·39-s + 0.312·41-s − 2.28·43-s + 1.89·47-s − 9/7·49-s − 0.140·51-s + 0.274·53-s − 0.264·57-s + 0.260·59-s + 1.79·61-s − 0.125·63-s + 0.244·67-s − 1.92·69-s − 0.356·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.103272306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.103272306\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - T + 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 7 T + 48 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T + 66 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 15 T + 138 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 13 T + 98 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 2 T + 90 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 102 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 14 T + 154 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 118 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T + 106 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
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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
−8.997528957183380856639128547220, −8.793650561668419680573362193994, −8.168943281239410472664575675721, −8.048758047369631093668196831019, −7.46648037751258757666665133777, −7.15385924270660454118239296500, −6.65027291572824663476297827528, −6.64181302254671328058020598098, −5.86583184397688393395089167191, −5.39909542629266022676540490754, −5.16370719923948546191761434699, −4.91312038638984344929673998378, −4.58463839275127112548637426849, −3.76833883289924275742149538885, −3.21427274172353820320320041477, −3.11515556066587340566824709190, −2.32204457559640186859699973925, −1.93382358463789597737631624704, −0.975106719384653365474296275911, −0.62056233407589325672839869313,
0.62056233407589325672839869313, 0.975106719384653365474296275911, 1.93382358463789597737631624704, 2.32204457559640186859699973925, 3.11515556066587340566824709190, 3.21427274172353820320320041477, 3.76833883289924275742149538885, 4.58463839275127112548637426849, 4.91312038638984344929673998378, 5.16370719923948546191761434699, 5.39909542629266022676540490754, 5.86583184397688393395089167191, 6.64181302254671328058020598098, 6.65027291572824663476297827528, 7.15385924270660454118239296500, 7.46648037751258757666665133777, 8.048758047369631093668196831019, 8.168943281239410472664575675721, 8.793650561668419680573362193994, 8.997528957183380856639128547220
