| L(s) = 1 | + 3-s − 7-s + 9-s − 2·19-s − 21-s − 25-s + 27-s + 2·31-s − 2·37-s + 49-s − 2·57-s − 63-s − 75-s + 81-s + 2·93-s + 2·103-s + 2·109-s − 2·111-s + ⋯ |
| L(s) = 1 | + 3-s − 7-s + 9-s − 2·19-s − 21-s − 25-s + 27-s + 2·31-s − 2·37-s + 49-s − 2·57-s − 63-s − 75-s + 81-s + 2·93-s + 2·103-s + 2·109-s − 2·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9771325928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9771325928\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( ( 1 + T )^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )^{2} \) |
| 37 | \( ( 1 + T )^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.00248601576233494248210963571, −10.51741367003951297414288068806, −9.933095829384933491368328763831, −8.900645363056699248590249829895, −8.208251127852839820786726173906, −6.99522497436652150927908471168, −6.17473861250043325926784560518, −4.47620797114688351993581425875, −3.43835064431643300987688001897, −2.21065757708639366344425174966,
2.21065757708639366344425174966, 3.43835064431643300987688001897, 4.47620797114688351993581425875, 6.17473861250043325926784560518, 6.99522497436652150927908471168, 8.208251127852839820786726173906, 8.900645363056699248590249829895, 9.933095829384933491368328763831, 10.51741367003951297414288068806, 12.00248601576233494248210963571
