| L(s) = 1 | + 4·2-s + 8·4-s + 4·5-s − 6·7-s + 8·8-s − 9-s + 16·10-s + 6·13-s − 24·14-s − 4·16-s − 4·18-s + 32·20-s + 11·25-s + 24·26-s − 48·28-s − 32·32-s − 24·35-s − 8·36-s − 6·37-s + 32·40-s − 4·45-s − 16·47-s + 13·49-s + 44·50-s + 48·52-s − 48·56-s − 6·61-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s + 1.78·5-s − 2.26·7-s + 2.82·8-s − 1/3·9-s + 5.05·10-s + 1.66·13-s − 6.41·14-s − 16-s − 0.942·18-s + 7.15·20-s + 11/5·25-s + 4.70·26-s − 9.07·28-s − 5.65·32-s − 4.05·35-s − 4/3·36-s − 0.986·37-s + 5.05·40-s − 0.596·45-s − 2.33·47-s + 13/7·49-s + 6.22·50-s + 6.65·52-s − 6.41·56-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.268228478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.268228478\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 57 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 117 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 177 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.05339485371485924951721421283, −12.76516899384898807864752740249, −12.20736739548103568740661783218, −11.56590017545580684504009540889, −10.99352238501862771238422594548, −10.51273807189150017031227137057, −9.604610484155876757013790124493, −9.582773517541079645706381718497, −8.961688636267927504156376033996, −8.320176015559696503119692646914, −6.81041947497301234744225777771, −6.60004458525053946393473006563, −6.16788341705015149443302022878, −6.03904336151484199706547834268, −5.21187206796291029661184384559, −4.97410276082022526443411931276, −3.61913858463703119456521164771, −3.60236536617269644391244940910, −2.92994329873548240638233866068, −2.13285056855256934939378264115,
2.13285056855256934939378264115, 2.92994329873548240638233866068, 3.60236536617269644391244940910, 3.61913858463703119456521164771, 4.97410276082022526443411931276, 5.21187206796291029661184384559, 6.03904336151484199706547834268, 6.16788341705015149443302022878, 6.60004458525053946393473006563, 6.81041947497301234744225777771, 8.320176015559696503119692646914, 8.961688636267927504156376033996, 9.582773517541079645706381718497, 9.604610484155876757013790124493, 10.51273807189150017031227137057, 10.99352238501862771238422594548, 11.56590017545580684504009540889, 12.20736739548103568740661783218, 12.76516899384898807864752740249, 13.05339485371485924951721421283
