| L(s) = 1 | + 2.67·2-s − 3-s + 5.17·4-s − 3.63·5-s − 2.67·6-s + 0.234·7-s + 8.50·8-s + 9-s − 9.72·10-s + 1.70·11-s − 5.17·12-s − 6.38·13-s + 0.628·14-s + 3.63·15-s + 12.4·16-s + 17-s + 2.67·18-s − 2.30·19-s − 18.7·20-s − 0.234·21-s + 4.56·22-s − 3.32·23-s − 8.50·24-s + 8.19·25-s − 17.1·26-s − 27-s + 1.21·28-s + ⋯ |
| L(s) = 1 | + 1.89·2-s − 0.577·3-s + 2.58·4-s − 1.62·5-s − 1.09·6-s + 0.0887·7-s + 3.00·8-s + 0.333·9-s − 3.07·10-s + 0.513·11-s − 1.49·12-s − 1.77·13-s + 0.168·14-s + 0.937·15-s + 3.10·16-s + 0.242·17-s + 0.631·18-s − 0.529·19-s − 4.20·20-s − 0.0512·21-s + 0.972·22-s − 0.693·23-s − 1.73·24-s + 1.63·25-s − 3.35·26-s − 0.192·27-s + 0.229·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 5 | \( 1 + 3.63T + 5T^{2} \) |
| 7 | \( 1 - 0.234T + 7T^{2} \) |
| 11 | \( 1 - 1.70T + 11T^{2} \) |
| 13 | \( 1 + 6.38T + 13T^{2} \) |
| 19 | \( 1 + 2.30T + 19T^{2} \) |
| 23 | \( 1 + 3.32T + 23T^{2} \) |
| 29 | \( 1 + 7.31T + 29T^{2} \) |
| 31 | \( 1 - 2.85T + 31T^{2} \) |
| 37 | \( 1 + 9.55T + 37T^{2} \) |
| 41 | \( 1 + 9.92T + 41T^{2} \) |
| 43 | \( 1 + 3.79T + 43T^{2} \) |
| 47 | \( 1 - 6.31T + 47T^{2} \) |
| 53 | \( 1 - 8.56T + 53T^{2} \) |
| 59 | \( 1 + 1.34T + 59T^{2} \) |
| 61 | \( 1 - 8.62T + 61T^{2} \) |
| 67 | \( 1 - 15.1T + 67T^{2} \) |
| 71 | \( 1 - 4.47T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 83 | \( 1 + 7.82T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 8.93T + 97T^{2} \) |
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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
−7.67295508831019639253186596451, −6.98530871928547048635504171899, −6.70442505287039296454005575316, −5.42697883085037602076063274388, −5.08277042060092997111272302475, −4.12527734640346734692929843598, −3.87410678797136017032322815139, −2.87658086121014872335695420236, −1.80857312064071776748798241522, 0,
1.80857312064071776748798241522, 2.87658086121014872335695420236, 3.87410678797136017032322815139, 4.12527734640346734692929843598, 5.08277042060092997111272302475, 5.42697883085037602076063274388, 6.70442505287039296454005575316, 6.98530871928547048635504171899, 7.67295508831019639253186596451
