L(s) = 1 | − 2·7-s + 7·13-s + 8·19-s + 5·25-s + 22·31-s − 2·37-s − 5·43-s − 11·49-s + 13·61-s − 5·67-s + 7·73-s + 13·79-s − 14·91-s − 14·97-s − 26·103-s − 2·109-s − 22·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
L(s) = 1 | − 0.755·7-s + 1.94·13-s + 1.83·19-s + 25-s + 3.95·31-s − 0.328·37-s − 0.762·43-s − 1.57·49-s + 1.66·61-s − 0.610·67-s + 0.819·73-s + 1.46·79-s − 1.46·91-s − 1.42·97-s − 2.56·103-s − 0.191·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 1.38·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 467856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.210440835\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.210440835\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.58879529167237870064069907125, −10.29493081965397001916050512431, −9.738899619383556626914400116752, −9.548448162232928792631016333427, −9.002651724365164864793080192360, −8.451295298584189737520893350974, −8.009888937858998171785173582196, −8.006174992979102293291042449070, −6.86417724246588241096283935443, −6.77188411045087413249770336375, −6.31109773730481798330143354615, −5.92610215535546317499030056157, −5.05159542177268024693787578947, −5.01504930672454926815802982300, −4.03767499036298439738601663654, −3.66299249576563919500134900564, −2.92436734698993218876634694339, −2.78396680113423007614359943982, −1.38182999948179400056795663445, −0.931201664825024208370380336841,
0.931201664825024208370380336841, 1.38182999948179400056795663445, 2.78396680113423007614359943982, 2.92436734698993218876634694339, 3.66299249576563919500134900564, 4.03767499036298439738601663654, 5.01504930672454926815802982300, 5.05159542177268024693787578947, 5.92610215535546317499030056157, 6.31109773730481798330143354615, 6.77188411045087413249770336375, 6.86417724246588241096283935443, 8.006174992979102293291042449070, 8.009888937858998171785173582196, 8.451295298584189737520893350974, 9.002651724365164864793080192360, 9.548448162232928792631016333427, 9.738899619383556626914400116752, 10.29493081965397001916050512431, 10.58879529167237870064069907125