| L(s) = 1 | + 2·3-s + 3·9-s − 12·17-s − 8·19-s − 10·25-s + 4·27-s − 20·31-s + 2·49-s − 24·51-s − 16·57-s + 24·59-s − 20·61-s + 8·67-s − 24·71-s − 4·73-s − 20·75-s + 20·79-s + 5·81-s − 40·93-s − 4·103-s + 24·107-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 4·147-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s − 2.91·17-s − 1.83·19-s − 2·25-s + 0.769·27-s − 3.59·31-s + 2/7·49-s − 3.36·51-s − 2.11·57-s + 3.12·59-s − 2.56·61-s + 0.977·67-s − 2.84·71-s − 0.468·73-s − 2.30·75-s + 2.25·79-s + 5/9·81-s − 4.14·93-s − 0.394·103-s + 2.32·107-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.329·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13307904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13307904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 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
−8.538399766442531984219396811998, −8.022966551071652002128017414209, −7.59707759520397959971622876271, −7.31951677011424166680877438000, −6.95568579887652239171870234778, −6.62511374681674227027721865582, −6.03602981822963843638243907407, −5.97097068620362093812913924997, −5.28874224654207232556139965524, −4.79034304675084481293124474546, −4.26680457765927646285070504421, −4.18395149966952732923383898861, −3.57711642746156448329744660044, −3.48865411638570202014117492821, −2.41265701346554372595535395820, −2.24714436017374929929764035610, −2.04842375875033097525944898377, −1.48810702238074128114328619340, 0, 0,
1.48810702238074128114328619340, 2.04842375875033097525944898377, 2.24714436017374929929764035610, 2.41265701346554372595535395820, 3.48865411638570202014117492821, 3.57711642746156448329744660044, 4.18395149966952732923383898861, 4.26680457765927646285070504421, 4.79034304675084481293124474546, 5.28874224654207232556139965524, 5.97097068620362093812913924997, 6.03602981822963843638243907407, 6.62511374681674227027721865582, 6.95568579887652239171870234778, 7.31951677011424166680877438000, 7.59707759520397959971622876271, 8.022966551071652002128017414209, 8.538399766442531984219396811998
