| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s + 9-s − 10-s − 7·13-s + 16-s − 3·17-s + 18-s − 20-s − 4·25-s − 7·26-s + 8·29-s + 32-s − 3·34-s + 36-s − 15·37-s − 40-s − 3·41-s − 45-s + 8·49-s − 4·50-s − 7·52-s + 16·53-s + 8·58-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.94·13-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.223·20-s − 4/5·25-s − 1.37·26-s + 1.48·29-s + 0.176·32-s − 0.514·34-s + 1/6·36-s − 2.46·37-s − 0.158·40-s − 0.468·41-s − 0.149·45-s + 8/7·49-s − 0.565·50-s − 0.970·52-s + 2.19·53-s + 1.05·58-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.055501009\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.055501009\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 63 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
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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
−7.81154328635533187538524199467, −7.51119765214149233644965187177, −7.11489322671850169361309124505, −6.65784920477251841733482820107, −6.49646548063570041719005511611, −5.55164487684975562564951164013, −5.32860271906811914933977559241, −4.89709917121813624987578840074, −4.31581315728837697906018762464, −4.07248787010848522106935186675, −3.39613297349508466029554013107, −2.81335820482034093857198935712, −2.25968138681366974796886304697, −1.77877039588945083399180211083, −0.54286079715813563443032978943,
0.54286079715813563443032978943, 1.77877039588945083399180211083, 2.25968138681366974796886304697, 2.81335820482034093857198935712, 3.39613297349508466029554013107, 4.07248787010848522106935186675, 4.31581315728837697906018762464, 4.89709917121813624987578840074, 5.32860271906811914933977559241, 5.55164487684975562564951164013, 6.49646548063570041719005511611, 6.65784920477251841733482820107, 7.11489322671850169361309124505, 7.51119765214149233644965187177, 7.81154328635533187538524199467
