| L(s) = 1 | + 3·4-s − 9-s + 2·11-s + 5·16-s + 12·29-s − 16·31-s − 3·36-s − 4·41-s + 6·44-s − 2·49-s + 8·59-s + 12·61-s + 3·64-s + 8·79-s + 81-s + 12·89-s − 2·99-s + 4·101-s + 4·109-s + 36·116-s + 3·121-s − 48·124-s + 127-s + 131-s + 137-s + 139-s − 5·144-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1/3·9-s + 0.603·11-s + 5/4·16-s + 2.22·29-s − 2.87·31-s − 1/2·36-s − 0.624·41-s + 0.904·44-s − 2/7·49-s + 1.04·59-s + 1.53·61-s + 3/8·64-s + 0.900·79-s + 1/9·81-s + 1.27·89-s − 0.201·99-s + 0.398·101-s + 0.383·109-s + 3.34·116-s + 3/11·121-s − 4.31·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.416·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.013204906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.013204906\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 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
−10.59894708919633338728836753681, −10.19827260779541492414597944516, −9.555443656409105116442124256140, −9.216227997030175190811964392102, −8.731712903125441223769712013187, −8.234062251810906359382240114906, −7.902016678713036797508885615016, −7.28333173292208514000135672280, −6.92888766118018441161434480736, −6.62075853550757221784196120429, −6.25940605281211111235736647426, −5.50583698078919530729031551332, −5.41211465666534224643436194700, −4.58884225367745366365122857165, −3.98217339862346779260223284124, −3.33165441182062053373742244392, −3.01455761281709906487139113814, −2.12216472826106110222686091145, −1.88885768745628063733273141121, −0.856711915194970839414152836025,
0.856711915194970839414152836025, 1.88885768745628063733273141121, 2.12216472826106110222686091145, 3.01455761281709906487139113814, 3.33165441182062053373742244392, 3.98217339862346779260223284124, 4.58884225367745366365122857165, 5.41211465666534224643436194700, 5.50583698078919530729031551332, 6.25940605281211111235736647426, 6.62075853550757221784196120429, 6.92888766118018441161434480736, 7.28333173292208514000135672280, 7.902016678713036797508885615016, 8.234062251810906359382240114906, 8.731712903125441223769712013187, 9.216227997030175190811964392102, 9.555443656409105116442124256140, 10.19827260779541492414597944516, 10.59894708919633338728836753681
