| L(s) = 1 | − 9-s + 8·11-s − 10·13-s + 8·23-s + 25-s − 2·29-s − 2·41-s + 8·43-s − 10·49-s − 2·73-s + 16·79-s − 8·81-s + 8·83-s − 8·99-s + 4·101-s + 8·103-s − 8·107-s + 10·117-s + 30·121-s + 127-s + 131-s + 137-s + 139-s − 80·143-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.41·11-s − 2.77·13-s + 1.66·23-s + 1/5·25-s − 0.371·29-s − 0.312·41-s + 1.21·43-s − 1.42·49-s − 0.234·73-s + 1.80·79-s − 8/9·81-s + 0.878·83-s − 0.804·99-s + 0.398·101-s + 0.788·103-s − 0.773·107-s + 0.924·117-s + 2.72·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 6.68·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 846400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.855775586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855775586\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 23 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
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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
−8.170956353412900815923955986835, −7.64040280897448541549723148092, −7.31206175251745513357589702566, −6.83971848444763701063488046455, −6.59004289166162230128427008184, −6.10658068462623892636562541093, −5.28692365042192612147005200827, −5.10441380998326312481409771124, −4.47915713957727601640290965833, −4.14484765468319742886790910786, −3.37774059573894264404118274190, −2.93434689064696478390688273458, −2.25807746961041685716103293995, −1.58835027542697041218002644564, −0.66422176411362324112839748951,
0.66422176411362324112839748951, 1.58835027542697041218002644564, 2.25807746961041685716103293995, 2.93434689064696478390688273458, 3.37774059573894264404118274190, 4.14484765468319742886790910786, 4.47915713957727601640290965833, 5.10441380998326312481409771124, 5.28692365042192612147005200827, 6.10658068462623892636562541093, 6.59004289166162230128427008184, 6.83971848444763701063488046455, 7.31206175251745513357589702566, 7.64040280897448541549723148092, 8.170956353412900815923955986835
