| L(s) = 1 | + 2·3-s + 3·9-s − 2·11-s + 7·17-s − 5·19-s − 2·25-s + 4·27-s − 4·33-s + 17·41-s − 43-s + 10·49-s + 14·51-s − 10·57-s − 14·59-s + 67-s + 14·73-s − 4·75-s + 5·81-s + 13·83-s − 17·89-s + 4·97-s − 6·99-s + 8·107-s − 6·113-s − 15·121-s + 34·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 0.603·11-s + 1.69·17-s − 1.14·19-s − 2/5·25-s + 0.769·27-s − 0.696·33-s + 2.65·41-s − 0.152·43-s + 10/7·49-s + 1.96·51-s − 1.32·57-s − 1.82·59-s + 0.122·67-s + 1.63·73-s − 0.461·75-s + 5/9·81-s + 1.42·83-s − 1.80·89-s + 0.406·97-s − 0.603·99-s + 0.773·107-s − 0.564·113-s − 1.36·121-s + 3.06·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2663424 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.656152093\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.656152093\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 28 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
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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
−7.67522783006779598715264778939, −7.42956161416208400828433613638, −6.92124659179671652343215865356, −6.33910797867418413143968235755, −5.90860756345739590165022049278, −5.58979315437410204037701804892, −4.99933346438639515361136041647, −4.45037463036239191273363613221, −4.09300436001783505651255754466, −3.56723309320935844961741029124, −3.14281599851705158477970898176, −2.52736156811944811604015468663, −2.24203397283754765033871921884, −1.46876605103055241600494574378, −0.70682897544927095541429698031,
0.70682897544927095541429698031, 1.46876605103055241600494574378, 2.24203397283754765033871921884, 2.52736156811944811604015468663, 3.14281599851705158477970898176, 3.56723309320935844961741029124, 4.09300436001783505651255754466, 4.45037463036239191273363613221, 4.99933346438639515361136041647, 5.58979315437410204037701804892, 5.90860756345739590165022049278, 6.33910797867418413143968235755, 6.92124659179671652343215865356, 7.42956161416208400828433613638, 7.67522783006779598715264778939
