L(s) = 1 | + 2-s + 4-s − 2·5-s + 8-s − 9-s − 2·10-s − 2·13-s + 16-s − 18-s − 2·20-s + 3·25-s − 2·26-s + 8·29-s + 32-s − 36-s − 4·37-s − 2·40-s + 8·41-s + 2·45-s + 2·49-s + 3·50-s − 2·52-s − 8·53-s + 8·58-s − 12·61-s + 64-s + 4·65-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.894·5-s + 0.353·8-s − 1/3·9-s − 0.632·10-s − 0.554·13-s + 1/4·16-s − 0.235·18-s − 0.447·20-s + 3/5·25-s − 0.392·26-s + 1.48·29-s + 0.176·32-s − 1/6·36-s − 0.657·37-s − 0.316·40-s + 1.24·41-s + 0.298·45-s + 2/7·49-s + 0.424·50-s − 0.277·52-s − 1.09·53-s + 1.05·58-s − 1.53·61-s + 1/8·64-s + 0.496·65-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
\(L(1)\) |
\(\approx\) |
\(2.233578078\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.233578078\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−7.974329083861169461711204688509, −7.60596310537835363074242785647, −7.09520691033161893312951010813, −6.71766181336806969605447659541, −6.24686353651114259833355695102, −5.83547941932160596416963661520, −5.23079364960260719714520326457, −4.77585675339976825744446972892, −4.47476552731699880082383495806, −3.92876495927149337247602505891, −3.36025638556076581595314614050, −2.90534186805765197938165483521, −2.41815729124572029272394317986, −1.57105541181048644388291882397, −0.59545077410238119269402403979,
0.59545077410238119269402403979, 1.57105541181048644388291882397, 2.41815729124572029272394317986, 2.90534186805765197938165483521, 3.36025638556076581595314614050, 3.92876495927149337247602505891, 4.47476552731699880082383495806, 4.77585675339976825744446972892, 5.23079364960260719714520326457, 5.83547941932160596416963661520, 6.24686353651114259833355695102, 6.71766181336806969605447659541, 7.09520691033161893312951010813, 7.60596310537835363074242785647, 7.974329083861169461711204688509