| L(s) = 1 | + 2.31·2-s + 3-s + 3.37·4-s + 0.731·5-s + 2.31·6-s − 2.06·7-s + 3.18·8-s + 9-s + 1.69·10-s − 4.53·11-s + 3.37·12-s − 5.27·13-s − 4.79·14-s + 0.731·15-s + 0.639·16-s − 17-s + 2.31·18-s + 4.22·19-s + 2.46·20-s − 2.06·21-s − 10.5·22-s + 1.41·23-s + 3.18·24-s − 4.46·25-s − 12.2·26-s + 27-s − 6.97·28-s + ⋯ |
| L(s) = 1 | + 1.63·2-s + 0.577·3-s + 1.68·4-s + 0.327·5-s + 0.946·6-s − 0.781·7-s + 1.12·8-s + 0.333·9-s + 0.536·10-s − 1.36·11-s + 0.974·12-s − 1.46·13-s − 1.28·14-s + 0.188·15-s + 0.159·16-s − 0.242·17-s + 0.546·18-s + 0.970·19-s + 0.552·20-s − 0.451·21-s − 2.24·22-s + 0.295·23-s + 0.650·24-s − 0.892·25-s − 2.39·26-s + 0.192·27-s − 1.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 - T \) |
| good | 2 | \( 1 - 2.31T + 2T^{2} \) |
| 5 | \( 1 - 0.731T + 5T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 11 | \( 1 + 4.53T + 11T^{2} \) |
| 13 | \( 1 + 5.27T + 13T^{2} \) |
| 19 | \( 1 - 4.22T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 3.67T + 29T^{2} \) |
| 31 | \( 1 + 5.47T + 31T^{2} \) |
| 37 | \( 1 + 6.37T + 37T^{2} \) |
| 41 | \( 1 - 5.71T + 41T^{2} \) |
| 43 | \( 1 + 7.14T + 43T^{2} \) |
| 47 | \( 1 - 7.61T + 47T^{2} \) |
| 53 | \( 1 - 9.76T + 53T^{2} \) |
| 59 | \( 1 + 4.22T + 59T^{2} \) |
| 61 | \( 1 + 7.97T + 61T^{2} \) |
| 67 | \( 1 + 0.501T + 67T^{2} \) |
| 71 | \( 1 - 7.26T + 71T^{2} \) |
| 73 | \( 1 + 4.76T + 73T^{2} \) |
| 79 | \( 1 + 15.7T + 79T^{2} \) |
| 83 | \( 1 + 8.98T + 83T^{2} \) |
| 89 | \( 1 - 5.58T + 89T^{2} \) |
| 97 | \( 1 + 7.11T + 97T^{2} \) |
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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
−7.27004933135900648550176571133, −6.78250311462090506666257270062, −5.76553793813886815995393656766, −5.36034650223284307130036886523, −4.70953858873447740972429539623, −3.90382108710026632951664400122, −3.00261831726748386217574622232, −2.71192355233127980510969125599, −1.88828911613691814616385902589, 0,
1.88828911613691814616385902589, 2.71192355233127980510969125599, 3.00261831726748386217574622232, 3.90382108710026632951664400122, 4.70953858873447740972429539623, 5.36034650223284307130036886523, 5.76553793813886815995393656766, 6.78250311462090506666257270062, 7.27004933135900648550176571133
