| L(s) = 1 | + 5·7-s + 7·13-s − 8·19-s − 10·25-s − 11·31-s + 37-s + 13·43-s + 18·49-s + 61-s − 11·67-s + 10·73-s + 13·79-s + 35·91-s + 19·97-s − 14·103-s + 19·109-s − 22·121-s + 127-s + 131-s − 40·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1.88·7-s + 1.94·13-s − 1.83·19-s − 2·25-s − 1.97·31-s + 0.164·37-s + 1.98·43-s + 18/7·49-s + 0.128·61-s − 1.34·67-s + 1.17·73-s + 1.46·79-s + 3.66·91-s + 1.92·97-s − 1.37·103-s + 1.81·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 3.46·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.083406252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.083406252\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−8.902678648353312107301548436596, −8.855654184481030745096709376121, −8.570818511531859968699547558832, −7.911638068117581806348353716587, −7.70251209612438451623131648381, −7.64143415997280399048965801694, −6.89493678956049389491769496479, −6.34214905477852995336998179975, −6.05583712084885767481934208477, −5.75500431826816915165607452392, −5.24144203248153702615291336161, −4.90005578062616256892419122176, −4.07604862199340948940615547105, −4.05127395161586616054015201291, −3.82790381884431799385968421150, −2.94733933372954155163963973555, −2.07593785656695397395522205076, −1.98144627595263414713135275594, −1.44263098814655918152925038770, −0.61604625888495184504997621295,
0.61604625888495184504997621295, 1.44263098814655918152925038770, 1.98144627595263414713135275594, 2.07593785656695397395522205076, 2.94733933372954155163963973555, 3.82790381884431799385968421150, 4.05127395161586616054015201291, 4.07604862199340948940615547105, 4.90005578062616256892419122176, 5.24144203248153702615291336161, 5.75500431826816915165607452392, 6.05583712084885767481934208477, 6.34214905477852995336998179975, 6.89493678956049389491769496479, 7.64143415997280399048965801694, 7.70251209612438451623131648381, 7.911638068117581806348353716587, 8.570818511531859968699547558832, 8.855654184481030745096709376121, 8.902678648353312107301548436596
