| L(s) = 1 | − 9-s − 2·11-s + 16·19-s + 12·29-s − 20·41-s − 2·49-s − 8·59-s − 4·61-s − 16·71-s + 8·79-s + 81-s + 12·89-s + 2·99-s + 4·101-s − 12·109-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s − 16·171-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 0.603·11-s + 3.67·19-s + 2.22·29-s − 3.12·41-s − 2/7·49-s − 1.04·59-s − 0.512·61-s − 1.89·71-s + 0.900·79-s + 1/9·81-s + 1.27·89-s + 0.201·99-s + 0.398·101-s − 1.14·109-s + 3/11·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s − 1.22·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.748273891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.748273891\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.189895042921327216937712360855, −7.65013478411981669343286991700, −7.63346943361923356497698039418, −7.16790271110017553532003256129, −6.73579524257721472130114487868, −6.55932841480255888887304010300, −5.98904602418638324469940649485, −5.59417785446494961163652515086, −5.37559866717523881095622124380, −4.85794366061380761291505487445, −4.83360876791575617152765891508, −4.32751423961662575011243128347, −3.54733043450864466714919115250, −3.25660479273900506999309294066, −3.05210487621140954362395056101, −2.78840941093918128962596917861, −1.99033277102822886095514260343, −1.51709663962674736945583071088, −1.02953379901027700480141895230, −0.46968592589364323727335755822,
0.46968592589364323727335755822, 1.02953379901027700480141895230, 1.51709663962674736945583071088, 1.99033277102822886095514260343, 2.78840941093918128962596917861, 3.05210487621140954362395056101, 3.25660479273900506999309294066, 3.54733043450864466714919115250, 4.32751423961662575011243128347, 4.83360876791575617152765891508, 4.85794366061380761291505487445, 5.37559866717523881095622124380, 5.59417785446494961163652515086, 5.98904602418638324469940649485, 6.55932841480255888887304010300, 6.73579524257721472130114487868, 7.16790271110017553532003256129, 7.63346943361923356497698039418, 7.65013478411981669343286991700, 8.189895042921327216937712360855
