| L(s) = 1 | + 2·3-s + 4·5-s + 3·9-s + 8·15-s − 8·23-s + 2·25-s + 4·27-s − 16·31-s − 20·37-s + 12·45-s − 16·47-s + 49-s + 12·53-s + 8·59-s + 24·67-s − 16·69-s + 8·71-s + 4·75-s + 5·81-s + 12·89-s − 32·93-s + 20·97-s − 40·111-s − 28·113-s − 32·115-s − 11·121-s − 28·125-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 9-s + 2.06·15-s − 1.66·23-s + 2/5·25-s + 0.769·27-s − 2.87·31-s − 3.28·37-s + 1.78·45-s − 2.33·47-s + 1/7·49-s + 1.64·53-s + 1.04·59-s + 2.93·67-s − 1.92·69-s + 0.949·71-s + 0.461·75-s + 5/9·81-s + 1.27·89-s − 3.31·93-s + 2.03·97-s − 3.79·111-s − 2.63·113-s − 2.98·115-s − 121-s − 2.50·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3415104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3415104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−7.35653811838304932393623429283, −6.86573975539025229905898029503, −6.42136658293515809263418102895, −6.18287900832404807661428217465, −5.41807067094670761229863645645, −5.25019213638990204722565722077, −5.07252108538521589319110157600, −3.90820666302401882520500245752, −3.69478854082878580470787829226, −3.56921408081238054859871707771, −2.55256266755253863914783357450, −2.06515692945354055956624655608, −1.95860198383842967586361157419, −1.43997509206912042814006725043, 0,
1.43997509206912042814006725043, 1.95860198383842967586361157419, 2.06515692945354055956624655608, 2.55256266755253863914783357450, 3.56921408081238054859871707771, 3.69478854082878580470787829226, 3.90820666302401882520500245752, 5.07252108538521589319110157600, 5.25019213638990204722565722077, 5.41807067094670761229863645645, 6.18287900832404807661428217465, 6.42136658293515809263418102895, 6.86573975539025229905898029503, 7.35653811838304932393623429283
