| L(s) = 1 | + 2·5-s − 6·9-s − 8·11-s + 16·19-s − 25-s + 12·29-s + 16·31-s + 4·41-s − 12·45-s + 49-s − 16·55-s − 12·61-s − 16·71-s + 32·79-s + 27·81-s − 12·89-s + 32·95-s + 48·99-s + 4·101-s − 20·109-s + 26·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2·9-s − 2.41·11-s + 3.67·19-s − 1/5·25-s + 2.22·29-s + 2.87·31-s + 0.624·41-s − 1.78·45-s + 1/7·49-s − 2.15·55-s − 1.53·61-s − 1.89·71-s + 3.60·79-s + 3·81-s − 1.27·89-s + 3.28·95-s + 4.82·99-s + 0.398·101-s − 1.91·109-s + 2.36·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.368178000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.368178000\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.694440338392361408863185872440, −9.457349313423188294666539437989, −8.660773281175179322159496718137, −8.188420417684287628984093773695, −7.81362030580797675551267597140, −7.47489161542833731239859573103, −6.42262084813563967264199691637, −6.06292563014554710285489032478, −5.38565578310165680979646788554, −5.22316409435338364257345412913, −4.71019542661529810418780789156, −3.18975677352219866037498641634, −2.79183800612725741835263061431, −2.62755636242588070812542768623, −0.916171764469637391994006820348,
0.916171764469637391994006820348, 2.62755636242588070812542768623, 2.79183800612725741835263061431, 3.18975677352219866037498641634, 4.71019542661529810418780789156, 5.22316409435338364257345412913, 5.38565578310165680979646788554, 6.06292563014554710285489032478, 6.42262084813563967264199691637, 7.47489161542833731239859573103, 7.81362030580797675551267597140, 8.188420417684287628984093773695, 8.660773281175179322159496718137, 9.457349313423188294666539437989, 9.694440338392361408863185872440
