| L(s) = 1 | − 2.24·2-s + 3-s + 3.04·4-s − 2.44·5-s − 2.24·6-s − 0.911·7-s − 2.35·8-s + 9-s + 5.49·10-s − 11-s + 3.04·12-s + 2.85·13-s + 2.04·14-s − 2.44·15-s − 0.801·16-s + 1.85·17-s − 2.24·18-s + 19-s − 7.45·20-s − 0.911·21-s + 2.24·22-s − 7.49·23-s − 2.35·24-s + 0.978·25-s − 6.40·26-s + 27-s − 2.78·28-s + ⋯ |
| L(s) = 1 | − 1.58·2-s + 0.577·3-s + 1.52·4-s − 1.09·5-s − 0.917·6-s − 0.344·7-s − 0.833·8-s + 0.333·9-s + 1.73·10-s − 0.301·11-s + 0.880·12-s + 0.790·13-s + 0.547·14-s − 0.631·15-s − 0.200·16-s + 0.448·17-s − 0.529·18-s + 0.229·19-s − 1.66·20-s − 0.198·21-s + 0.479·22-s − 1.56·23-s − 0.481·24-s + 0.195·25-s − 1.25·26-s + 0.192·27-s − 0.525·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 5 | \( 1 + 2.44T + 5T^{2} \) |
| 7 | \( 1 + 0.911T + 7T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 23 | \( 1 + 7.49T + 23T^{2} \) |
| 29 | \( 1 + 9.78T + 29T^{2} \) |
| 31 | \( 1 - 3.78T + 31T^{2} \) |
| 37 | \( 1 + 8.38T + 37T^{2} \) |
| 41 | \( 1 - 5.96T + 41T^{2} \) |
| 43 | \( 1 - 3.48T + 43T^{2} \) |
| 47 | \( 1 + 3.57T + 47T^{2} \) |
| 53 | \( 1 + 9.26T + 53T^{2} \) |
| 59 | \( 1 - 2.02T + 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 9.05T + 71T^{2} \) |
| 73 | \( 1 - 0.917T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 4.00T + 89T^{2} \) |
| 97 | \( 1 - 4.40T + 97T^{2} \) |
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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
−9.940900509486699045950458012238, −9.254690122007701533168539136990, −8.313699061430499401122284942532, −7.85836576701250189587419678954, −7.16062138988863472952409263107, −5.96422856770458226141277004652, −4.21525308763110557170585096079, −3.21180224569036182839122700737, −1.69921336580005213243429379944, 0,
1.69921336580005213243429379944, 3.21180224569036182839122700737, 4.21525308763110557170585096079, 5.96422856770458226141277004652, 7.16062138988863472952409263107, 7.85836576701250189587419678954, 8.313699061430499401122284942532, 9.254690122007701533168539136990, 9.940900509486699045950458012238
