| L(s) = 1 | − 2.52·2-s − 3-s + 4.35·4-s − 1.87·5-s + 2.52·6-s − 3.72·7-s − 5.92·8-s + 9-s + 4.72·10-s − 2.94·11-s − 4.35·12-s − 4.80·13-s + 9.37·14-s + 1.87·15-s + 6.24·16-s − 17-s − 2.52·18-s − 2.96·19-s − 8.15·20-s + 3.72·21-s + 7.42·22-s − 3.55·23-s + 5.92·24-s − 1.48·25-s + 12.1·26-s − 27-s − 16.1·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 0.577·3-s + 2.17·4-s − 0.838·5-s + 1.02·6-s − 1.40·7-s − 2.09·8-s + 0.333·9-s + 1.49·10-s − 0.888·11-s − 1.25·12-s − 1.33·13-s + 2.50·14-s + 0.484·15-s + 1.56·16-s − 0.242·17-s − 0.594·18-s − 0.680·19-s − 1.82·20-s + 0.812·21-s + 1.58·22-s − 0.741·23-s + 1.21·24-s − 0.297·25-s + 2.37·26-s − 0.192·27-s − 3.06·28-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 | \( 1 + 2.52T + 2T^{2} \) |
| 5 | \( 1 + 1.87T + 5T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 11 | \( 1 + 2.94T + 11T^{2} \) |
| 13 | \( 1 + 4.80T + 13T^{2} \) |
| 19 | \( 1 + 2.96T + 19T^{2} \) |
| 23 | \( 1 + 3.55T + 23T^{2} \) |
| 29 | \( 1 + 2.12T + 29T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 - 0.184T + 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 - 0.542T + 43T^{2} \) |
| 47 | \( 1 - 8.00T + 47T^{2} \) |
| 53 | \( 1 - 5.77T + 53T^{2} \) |
| 59 | \( 1 + 4.76T + 59T^{2} \) |
| 61 | \( 1 + 8.87T + 61T^{2} \) |
| 67 | \( 1 - 6.37T + 67T^{2} \) |
| 71 | \( 1 - 0.652T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 5.92T + 79T^{2} \) |
| 83 | \( 1 - 7.36T + 83T^{2} \) |
| 89 | \( 1 - 13.2T + 89T^{2} \) |
| 97 | \( 1 + 6.53T + 97T^{2} \) |
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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
−7.37290030943568439309086552855, −7.30847135544416957859637341782, −6.28386658870456411459072078525, −5.83446821729116962819690531606, −4.65747435267863403285727271843, −3.76074503444398907518467392778, −2.70632639687168502225349538745, −2.13152250082447210827293216964, −0.57305346918020445966960042984, 0,
0.57305346918020445966960042984, 2.13152250082447210827293216964, 2.70632639687168502225349538745, 3.76074503444398907518467392778, 4.65747435267863403285727271843, 5.83446821729116962819690531606, 6.28386658870456411459072078525, 7.30847135544416957859637341782, 7.37290030943568439309086552855
