| L(s) = 1 | + 2-s − 2·5-s + 7-s − 8-s − 2·10-s + 5·11-s + 12·13-s + 14-s − 16-s − 4·17-s + 4·19-s + 5·22-s − 4·23-s + 5·25-s + 12·26-s − 14·29-s − 3·31-s − 4·34-s − 2·35-s − 8·37-s + 4·38-s + 2·40-s + 12·41-s + 16·43-s − 4·46-s + 6·47-s − 6·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s + 1.50·11-s + 3.32·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.917·19-s + 1.06·22-s − 0.834·23-s + 25-s + 2.35·26-s − 2.59·29-s − 0.538·31-s − 0.685·34-s − 0.338·35-s − 1.31·37-s + 0.648·38-s + 0.316·40-s + 1.87·41-s + 2.43·43-s − 0.589·46-s + 0.875·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142884 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.379065075\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.379065075\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7 T - 10 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
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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
−11.45688228847364560918921616366, −11.23231644065076563021275437414, −10.83918973737132585995253991887, −10.70261601664224664085062975677, −9.488164441171811616303707442350, −9.071480386247876814458494402307, −8.980879058013000285634068643822, −8.475734715203401914282448175799, −7.67629939923188281731416952850, −7.55720783839625913117175217512, −6.58420840546510374209119620213, −6.39218317429906017235492956273, −5.67013401985455615137700811859, −5.48322240473819835554297886333, −4.22831925259862427095549030403, −4.06368059801025279216366444096, −3.77751641580697688276586904034, −3.16585513149263045912795950809, −1.84019554945691324312473519307, −1.07253666696515717695817810772,
1.07253666696515717695817810772, 1.84019554945691324312473519307, 3.16585513149263045912795950809, 3.77751641580697688276586904034, 4.06368059801025279216366444096, 4.22831925259862427095549030403, 5.48322240473819835554297886333, 5.67013401985455615137700811859, 6.39218317429906017235492956273, 6.58420840546510374209119620213, 7.55720783839625913117175217512, 7.67629939923188281731416952850, 8.475734715203401914282448175799, 8.980879058013000285634068643822, 9.071480386247876814458494402307, 9.488164441171811616303707442350, 10.70261601664224664085062975677, 10.83918973737132585995253991887, 11.23231644065076563021275437414, 11.45688228847364560918921616366
