| L(s) = 1 | − 4·7-s − 2·13-s + 19-s − 10·25-s + 7·31-s + 10·37-s − 5·43-s + 9·49-s + 61-s + 16·67-s − 17·73-s + 4·79-s + 8·91-s + 19·97-s + 40·103-s − 17·109-s − 22·121-s + 127-s + 131-s − 4·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.554·13-s + 0.229·19-s − 2·25-s + 1.25·31-s + 1.64·37-s − 0.762·43-s + 9/7·49-s + 0.128·61-s + 1.95·67-s − 1.98·73-s + 0.450·79-s + 0.838·91-s + 1.92·97-s + 3.94·103-s − 1.62·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s − 0.346·133-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5143824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.015417720\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.015417720\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
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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
−9.291122328611718349895599706417, −8.958011323934146674340728859508, −8.425267446671817411179928971512, −8.007623087298332746234245775671, −7.61218400205677995047146541298, −7.40088258895552779700720193226, −6.82927548833839094404464875102, −6.30817956212715710106039443967, −6.25246162473715572955443888923, −5.85281977791787056277457947149, −5.22556683155357210348381521104, −4.89847339124623733294036718634, −4.28581574043032567823460635352, −3.87201055401858139275699630868, −3.51342737769171320840106795590, −2.93978152154037080513733695321, −2.51000669199540901406570886824, −2.06653202313522153868653093689, −1.16353929674270346245792418559, −0.37259392282041021892127344074,
0.37259392282041021892127344074, 1.16353929674270346245792418559, 2.06653202313522153868653093689, 2.51000669199540901406570886824, 2.93978152154037080513733695321, 3.51342737769171320840106795590, 3.87201055401858139275699630868, 4.28581574043032567823460635352, 4.89847339124623733294036718634, 5.22556683155357210348381521104, 5.85281977791787056277457947149, 6.25246162473715572955443888923, 6.30817956212715710106039443967, 6.82927548833839094404464875102, 7.40088258895552779700720193226, 7.61218400205677995047146541298, 8.007623087298332746234245775671, 8.425267446671817411179928971512, 8.958011323934146674340728859508, 9.291122328611718349895599706417
