| L(s) = 1 | − 0.303·2-s + 1.58·3-s − 1.90·4-s − 2.73·5-s − 0.480·6-s + 1.58·7-s + 1.18·8-s − 0.499·9-s + 0.831·10-s − 11-s − 3.01·12-s + 0.112·13-s − 0.481·14-s − 4.33·15-s + 3.45·16-s − 0.774·17-s + 0.151·18-s + 4.98·19-s + 5.22·20-s + 2.50·21-s + 0.303·22-s + 1.87·24-s + 2.50·25-s − 0.0342·26-s − 5.53·27-s − 3.02·28-s − 0.203·29-s + ⋯ |
| L(s) = 1 | − 0.214·2-s + 0.912·3-s − 0.953·4-s − 1.22·5-s − 0.196·6-s + 0.599·7-s + 0.419·8-s − 0.166·9-s + 0.263·10-s − 0.301·11-s − 0.870·12-s + 0.0312·13-s − 0.128·14-s − 1.11·15-s + 0.863·16-s − 0.187·17-s + 0.0357·18-s + 1.14·19-s + 1.16·20-s + 0.547·21-s + 0.0647·22-s + 0.382·24-s + 0.500·25-s − 0.00670·26-s − 1.06·27-s − 0.571·28-s − 0.0377·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.180123641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.180123641\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 0.303T + 2T^{2} \) |
| 3 | \( 1 - 1.58T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 - 1.58T + 7T^{2} \) |
| 13 | \( 1 - 0.112T + 13T^{2} \) |
| 17 | \( 1 + 0.774T + 17T^{2} \) |
| 19 | \( 1 - 4.98T + 19T^{2} \) |
| 29 | \( 1 + 0.203T + 29T^{2} \) |
| 31 | \( 1 + 1.08T + 31T^{2} \) |
| 37 | \( 1 + 0.652T + 37T^{2} \) |
| 41 | \( 1 + 1.78T + 41T^{2} \) |
| 43 | \( 1 - 5.85T + 43T^{2} \) |
| 47 | \( 1 + 7.48T + 47T^{2} \) |
| 53 | \( 1 + 3.20T + 53T^{2} \) |
| 59 | \( 1 - 2.28T + 59T^{2} \) |
| 61 | \( 1 + 8.89T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 5.83T + 71T^{2} \) |
| 73 | \( 1 + 2.15T + 73T^{2} \) |
| 79 | \( 1 - 5.14T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 + 5.83T + 97T^{2} \) |
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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
−8.156129306162210315028704232675, −7.74610864474913837154647059238, −7.10360506645412756629902696946, −5.80468469928402572032193502570, −5.01387233329968131888458272008, −4.38130091569072930979799465498, −3.56955036252950093875942453238, −3.08739179941947458575689269915, −1.82607372016780097964638139317, −0.57094702878128841065593496465,
0.57094702878128841065593496465, 1.82607372016780097964638139317, 3.08739179941947458575689269915, 3.56955036252950093875942453238, 4.38130091569072930979799465498, 5.01387233329968131888458272008, 5.80468469928402572032193502570, 7.10360506645412756629902696946, 7.74610864474913837154647059238, 8.156129306162210315028704232675
