| L(s) = 1 | + 4-s − 3·16-s + 6·17-s + 12·23-s + 7·25-s − 6·29-s + 16·43-s + 14·49-s + 6·53-s + 2·61-s − 7·64-s + 6·68-s + 8·79-s + 12·92-s + 7·100-s + 6·101-s − 20·103-s − 12·107-s + 30·113-s − 6·116-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3/4·16-s + 1.45·17-s + 2.50·23-s + 7/5·25-s − 1.11·29-s + 2.43·43-s + 2·49-s + 0.824·53-s + 0.256·61-s − 7/8·64-s + 0.727·68-s + 0.900·79-s + 1.25·92-s + 7/10·100-s + 0.597·101-s − 1.97·103-s − 1.16·107-s + 2.82·113-s − 0.557·116-s + 2·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.242931511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.242931511\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
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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
−9.375096927701881705960205628574, −9.370728066328608374647877076727, −8.823474127516141877521569974711, −8.697687121352489327412843893925, −8.001683704352429602662919552260, −7.39048057250155899981034009447, −7.32819167552791801947553849196, −7.02074979626099691780913032993, −6.48196294115256074729197891516, −5.93343058081555144490635597267, −5.60133805062073089091655830069, −5.06432960319164524312675264118, −4.83901673305684160672794250111, −4.06177836976334938606106985442, −3.72777606986827122722357496565, −3.00443930622239499595071551820, −2.70159612058965043853668876459, −2.18759707835427310790097958301, −1.19872402468900595664002206459, −0.833056772828202747498937380587,
0.833056772828202747498937380587, 1.19872402468900595664002206459, 2.18759707835427310790097958301, 2.70159612058965043853668876459, 3.00443930622239499595071551820, 3.72777606986827122722357496565, 4.06177836976334938606106985442, 4.83901673305684160672794250111, 5.06432960319164524312675264118, 5.60133805062073089091655830069, 5.93343058081555144490635597267, 6.48196294115256074729197891516, 7.02074979626099691780913032993, 7.32819167552791801947553849196, 7.39048057250155899981034009447, 8.001683704352429602662919552260, 8.697687121352489327412843893925, 8.823474127516141877521569974711, 9.370728066328608374647877076727, 9.375096927701881705960205628574
