| L(s) = 1 | + 4·5-s − 3·9-s − 16·19-s + 2·25-s + 12·29-s + 8·43-s − 12·45-s + 16·47-s + 49-s + 12·53-s + 8·67-s + 16·71-s + 20·73-s + 9·81-s − 64·95-s − 12·97-s + 4·101-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 48·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 9-s − 3.67·19-s + 2/5·25-s + 2.22·29-s + 1.21·43-s − 1.78·45-s + 2.33·47-s + 1/7·49-s + 1.64·53-s + 0.977·67-s + 1.89·71-s + 2.34·73-s + 81-s − 6.56·95-s − 1.21·97-s + 0.398·101-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.152524652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.152524652\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−8.560607806232820418165457845354, −8.272203299109375537884659130698, −7.902596514850714806179317231692, −6.84383926850302326577754552348, −6.67949194679031572918581545842, −6.20669276463558444353009673367, −5.82613210142107824134972732669, −5.50140758689236696304146355325, −4.83297349612888492276603651901, −4.18465932715971087267449515138, −3.83348667018088193321396729321, −2.66451378084199854982327930143, −2.24055198785635289229441777506, −2.14556791802447766709825303506, −0.76540944404158607646425160976,
0.76540944404158607646425160976, 2.14556791802447766709825303506, 2.24055198785635289229441777506, 2.66451378084199854982327930143, 3.83348667018088193321396729321, 4.18465932715971087267449515138, 4.83297349612888492276603651901, 5.50140758689236696304146355325, 5.82613210142107824134972732669, 6.20669276463558444353009673367, 6.67949194679031572918581545842, 6.84383926850302326577754552348, 7.902596514850714806179317231692, 8.272203299109375537884659130698, 8.560607806232820418165457845354
