| L(s) = 1 | + 6·9-s − 16·17-s + 8·25-s + 16·41-s − 14·49-s − 12·73-s + 27·81-s + 20·89-s − 16·97-s + 28·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 96·153-s + 157-s + 163-s + 167-s − 24·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·9-s − 3.88·17-s + 8/5·25-s + 2.49·41-s − 2·49-s − 1.40·73-s + 3·81-s + 2.11·89-s − 1.62·97-s + 2.63·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 7.76·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.84·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1048576 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.044004972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.044004972\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−10.13217898193172963257822181326, −9.688603366656961642275273958425, −9.314310229915961414502384416886, −8.911885291963261747923394328740, −8.672764413395070798897038849820, −8.119433769299278459419774122301, −7.39682947434747951407306110318, −7.27373285977552829048019520380, −6.77561244584082396014582566263, −6.37080064595568255930670415626, −6.22609404552281807143423112678, −5.25359728058462776446139238738, −4.57312390188671317000784340696, −4.50874480093786555185495387665, −4.26740037117766695282554928634, −3.46492958112130918888057512261, −2.69889583061787347538234318137, −2.12820322088497769881084109262, −1.65334165339279152226966748706, −0.66015831564846254449894922385,
0.66015831564846254449894922385, 1.65334165339279152226966748706, 2.12820322088497769881084109262, 2.69889583061787347538234318137, 3.46492958112130918888057512261, 4.26740037117766695282554928634, 4.50874480093786555185495387665, 4.57312390188671317000784340696, 5.25359728058462776446139238738, 6.22609404552281807143423112678, 6.37080064595568255930670415626, 6.77561244584082396014582566263, 7.27373285977552829048019520380, 7.39682947434747951407306110318, 8.119433769299278459419774122301, 8.672764413395070798897038849820, 8.911885291963261747923394328740, 9.314310229915961414502384416886, 9.688603366656961642275273958425, 10.13217898193172963257822181326
