| L(s) = 1 | − 2.62·3-s + 0.864·7-s + 3.89·9-s − 2·11-s + 2.62·13-s − 17-s + 0.896·19-s − 2.27·21-s − 3.13·23-s − 2.35·27-s + 9.49·29-s − 9.01·31-s + 5.25·33-s − 10.1·37-s − 6.89·39-s + 9.52·41-s + 7.25·43-s − 10.4·47-s − 6.25·49-s + 2.62·51-s − 11.4·53-s − 2.35·57-s + 4.14·59-s + 3.28·61-s + 3.36·63-s + 1.25·67-s + 8.23·69-s + ⋯ |
| L(s) = 1 | − 1.51·3-s + 0.326·7-s + 1.29·9-s − 0.603·11-s + 0.728·13-s − 0.242·17-s + 0.205·19-s − 0.495·21-s − 0.653·23-s − 0.453·27-s + 1.76·29-s − 1.61·31-s + 0.914·33-s − 1.67·37-s − 1.10·39-s + 1.48·41-s + 1.10·43-s − 1.51·47-s − 0.893·49-s + 0.367·51-s − 1.56·53-s − 0.311·57-s + 0.540·59-s + 0.420·61-s + 0.424·63-s + 0.153·67-s + 0.991·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2.62T + 3T^{2} \) |
| 7 | \( 1 - 0.864T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.62T + 13T^{2} \) |
| 19 | \( 1 - 0.896T + 19T^{2} \) |
| 23 | \( 1 + 3.13T + 23T^{2} \) |
| 29 | \( 1 - 9.49T + 29T^{2} \) |
| 31 | \( 1 + 9.01T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 9.52T + 41T^{2} \) |
| 43 | \( 1 - 7.25T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 - 3.28T + 61T^{2} \) |
| 67 | \( 1 - 1.25T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 2.62T + 73T^{2} \) |
| 79 | \( 1 - 7.91T + 79T^{2} \) |
| 83 | \( 1 + 4.20T + 83T^{2} \) |
| 89 | \( 1 - 4.14T + 89T^{2} \) |
| 97 | \( 1 - 6.14T + 97T^{2} \) |
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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
−7.57849856463584748155002342804, −6.61623913568969077297545999502, −6.28410673292826046977199771131, −5.39324542703198490567782145223, −5.02718891836710386325051753479, −4.20123647921346129921930671706, −3.28801190214405179121242369150, −2.06920387046711438500754588862, −1.07006815035809038407822934719, 0,
1.07006815035809038407822934719, 2.06920387046711438500754588862, 3.28801190214405179121242369150, 4.20123647921346129921930671706, 5.02718891836710386325051753479, 5.39324542703198490567782145223, 6.28410673292826046977199771131, 6.61623913568969077297545999502, 7.57849856463584748155002342804
