| L(s) = 1 | − 2-s + 4-s + 3·5-s − 8-s − 2·9-s − 3·10-s + 13-s + 16-s − 3·17-s + 2·18-s + 3·20-s − 25-s − 26-s − 3·29-s − 32-s + 3·34-s − 2·36-s − 5·37-s − 3·40-s + 15·41-s − 6·45-s − 4·49-s + 50-s + 52-s + 6·53-s + 3·58-s − 11·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.353·8-s − 2/3·9-s − 0.948·10-s + 0.277·13-s + 1/4·16-s − 0.727·17-s + 0.471·18-s + 0.670·20-s − 1/5·25-s − 0.196·26-s − 0.557·29-s − 0.176·32-s + 0.514·34-s − 1/3·36-s − 0.821·37-s − 0.474·40-s + 2.34·41-s − 0.894·45-s − 4/7·49-s + 0.141·50-s + 0.138·52-s + 0.824·53-s + 0.393·58-s − 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5792 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5792 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7426017584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7426017584\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 + T \) |
| 181 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 20 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
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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
−12.00955446952566062516831963552, −11.25307076375712601034740297514, −10.95734444896584609375042595477, −10.24949577024123301812024667933, −9.689448936766190710462567174021, −9.173847103355220735547726566332, −8.747537823519097228078111103344, −7.965905602869126536586482827876, −7.27356970543258952991675186551, −6.40875296722595694451334519466, −5.91603355397174398209604505550, −5.34556505829101948517607491106, −4.12506025686418199543457614455, −2.81964959448269450069319776233, −1.87628743613496808761921596450,
1.87628743613496808761921596450, 2.81964959448269450069319776233, 4.12506025686418199543457614455, 5.34556505829101948517607491106, 5.91603355397174398209604505550, 6.40875296722595694451334519466, 7.27356970543258952991675186551, 7.965905602869126536586482827876, 8.747537823519097228078111103344, 9.173847103355220735547726566332, 9.689448936766190710462567174021, 10.24949577024123301812024667933, 10.95734444896584609375042595477, 11.25307076375712601034740297514, 12.00955446952566062516831963552
