| L(s) = 1 | + 1.79·2-s − 3-s + 1.20·4-s + 3·5-s − 1.79·6-s + 7-s − 1.41·8-s + 9-s + 5.37·10-s − 11-s − 1.20·12-s + 13-s + 1.79·14-s − 3·15-s − 4.95·16-s + 7.58·17-s + 1.79·18-s − 6.58·19-s + 3.62·20-s − 21-s − 1.79·22-s − 5.58·23-s + 1.41·24-s + 4·25-s + 1.79·26-s − 27-s + 1.20·28-s + ⋯ |
| L(s) = 1 | + 1.26·2-s − 0.577·3-s + 0.604·4-s + 1.34·5-s − 0.731·6-s + 0.377·7-s − 0.501·8-s + 0.333·9-s + 1.69·10-s − 0.301·11-s − 0.348·12-s + 0.277·13-s + 0.478·14-s − 0.774·15-s − 1.23·16-s + 1.83·17-s + 0.422·18-s − 1.51·19-s + 0.810·20-s − 0.218·21-s − 0.381·22-s − 1.16·23-s + 0.289·24-s + 0.800·25-s + 0.351·26-s − 0.192·27-s + 0.228·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.153673043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153673043\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 1.79T + 2T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 7.58T + 17T^{2} \) |
| 19 | \( 1 + 6.58T + 19T^{2} \) |
| 23 | \( 1 + 5.58T + 23T^{2} \) |
| 29 | \( 1 + 8.16T + 29T^{2} \) |
| 31 | \( 1 - 3.58T + 31T^{2} \) |
| 37 | \( 1 - T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 + 9.58T + 53T^{2} \) |
| 59 | \( 1 - 4.58T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 8.58T + 67T^{2} \) |
| 71 | \( 1 - 11.1T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 - 7.16T + 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 - 9.16T + 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−12.47963995589029819295526310648, −11.50530040163972206601299999110, −10.36938564096962006674759496066, −9.577806794640621249754897573339, −8.161939317897230989109662638355, −6.53380845063870952667786544858, −5.75369936884517779737088399372, −5.11749517832103761916153067282, −3.75271478128334122181597471475, −2.06016902653797192392454285505,
2.06016902653797192392454285505, 3.75271478128334122181597471475, 5.11749517832103761916153067282, 5.75369936884517779737088399372, 6.53380845063870952667786544858, 8.161939317897230989109662638355, 9.577806794640621249754897573339, 10.36938564096962006674759496066, 11.50530040163972206601299999110, 12.47963995589029819295526310648
