| L(s) = 1 | + 2·5-s − 25-s − 20·29-s + 20·41-s + 14·49-s − 20·61-s − 20·89-s − 4·101-s + 12·109-s − 22·121-s − 12·125-s + 127-s + 131-s + 137-s + 139-s − 40·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1/5·25-s − 3.71·29-s + 3.12·41-s + 2·49-s − 2.56·61-s − 2.11·89-s − 0.398·101-s + 1.14·109-s − 2·121-s − 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.049784379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.049784379\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−9.039523538675511724137548248194, −8.843930245042817580366110826752, −8.121301343405760959183482689883, −7.59812668664369178690132370819, −7.57528557064141829846046065116, −7.23977407670089953404899970467, −6.65663653968476003622678429574, −6.07654827596394856043136690629, −5.94202489716515719519305952189, −5.45548354361408309909675659107, −5.43672455959816009202069942142, −4.62227008353658746517349341601, −4.12352554763270927032121393482, −3.93645984749375776154401942758, −3.38710099032025461294869529499, −2.61014959165013437449921065583, −2.46928385096918822876953736359, −1.73915793170354511475675263602, −1.42488959138947030138475184456, −0.44478904099776668615566349508,
0.44478904099776668615566349508, 1.42488959138947030138475184456, 1.73915793170354511475675263602, 2.46928385096918822876953736359, 2.61014959165013437449921065583, 3.38710099032025461294869529499, 3.93645984749375776154401942758, 4.12352554763270927032121393482, 4.62227008353658746517349341601, 5.43672455959816009202069942142, 5.45548354361408309909675659107, 5.94202489716515719519305952189, 6.07654827596394856043136690629, 6.65663653968476003622678429574, 7.23977407670089953404899970467, 7.57528557064141829846046065116, 7.59812668664369178690132370819, 8.121301343405760959183482689883, 8.843930245042817580366110826752, 9.039523538675511724137548248194
