| L(s) = 1 | + 2·3-s − 5·9-s + 28·11-s − 50·25-s − 28·27-s + 56·33-s − 98·49-s + 164·59-s + 284·73-s − 100·75-s − 11·81-s + 316·83-s − 188·97-s − 140·99-s + 356·107-s + 346·121-s + 127-s + 131-s + 137-s + 139-s − 196·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 338·169-s + ⋯ |
| L(s) = 1 | + 2/3·3-s − 5/9·9-s + 2.54·11-s − 2·25-s − 1.03·27-s + 1.69·33-s − 2·49-s + 2.77·59-s + 3.89·73-s − 4/3·75-s − 0.135·81-s + 3.80·83-s − 1.93·97-s − 1.41·99-s + 3.32·107-s + 2.85·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 4/3·147-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147456 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.978335168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.978335168\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - 2 T + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )( 1 + 2 T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )( 1 + 34 T + p^{2} T^{2} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )( 1 + 62 T + p^{2} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 158 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )( 1 + 146 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{2} \) |
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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
−11.30055324486475863609284737102, −11.23527059280567985679956362223, −10.33138506087614028230797350441, −9.677001820301203809842832195341, −9.546970934611118927338874608792, −9.121667426761521819345436313131, −8.662613721105539014594154100082, −7.996752552438234226508357093334, −7.928489373145837548766333028808, −7.04323302708937753741280308066, −6.48160258826234289988760967008, −6.32963652806260335913268035065, −5.59102255421789313541249810055, −4.99198788769807268505217920982, −4.15431797147578082139720404995, −3.59643568049777889321244605669, −3.52472761402188817023236344179, −2.26753061296179472572078955387, −1.82411520959539345302716355820, −0.75509870246957107200116109617,
0.75509870246957107200116109617, 1.82411520959539345302716355820, 2.26753061296179472572078955387, 3.52472761402188817023236344179, 3.59643568049777889321244605669, 4.15431797147578082139720404995, 4.99198788769807268505217920982, 5.59102255421789313541249810055, 6.32963652806260335913268035065, 6.48160258826234289988760967008, 7.04323302708937753741280308066, 7.928489373145837548766333028808, 7.996752552438234226508357093334, 8.662613721105539014594154100082, 9.121667426761521819345436313131, 9.546970934611118927338874608792, 9.677001820301203809842832195341, 10.33138506087614028230797350441, 11.23527059280567985679956362223, 11.30055324486475863609284737102
