| L(s) = 1 | − 2-s − 0.466·3-s + 4-s − 3.38·5-s + 0.466·6-s − 7-s − 8-s − 2.78·9-s + 3.38·10-s + 0.822·11-s − 0.466·12-s + 14-s + 1.58·15-s + 16-s + 4.58·17-s + 2.78·18-s − 5.90·19-s − 3.38·20-s + 0.466·21-s − 0.822·22-s − 6.13·23-s + 0.466·24-s + 6.48·25-s + 2.69·27-s − 28-s − 6.86·29-s − 1.58·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.269·3-s + 0.5·4-s − 1.51·5-s + 0.190·6-s − 0.377·7-s − 0.353·8-s − 0.927·9-s + 1.07·10-s + 0.248·11-s − 0.134·12-s + 0.267·14-s + 0.408·15-s + 0.250·16-s + 1.11·17-s + 0.655·18-s − 1.35·19-s − 0.757·20-s + 0.101·21-s − 0.175·22-s − 1.27·23-s + 0.0952·24-s + 1.29·25-s + 0.519·27-s − 0.188·28-s − 1.27·29-s − 0.288·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2979975510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2979975510\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.466T + 3T^{2} \) |
| 5 | \( 1 + 3.38T + 5T^{2} \) |
| 11 | \( 1 - 0.822T + 11T^{2} \) |
| 17 | \( 1 - 4.58T + 17T^{2} \) |
| 19 | \( 1 + 5.90T + 19T^{2} \) |
| 23 | \( 1 + 6.13T + 23T^{2} \) |
| 29 | \( 1 + 6.86T + 29T^{2} \) |
| 31 | \( 1 + 4.28T + 31T^{2} \) |
| 37 | \( 1 + 9.69T + 37T^{2} \) |
| 41 | \( 1 - 0.0893T + 41T^{2} \) |
| 43 | \( 1 - 7.35T + 43T^{2} \) |
| 47 | \( 1 + 11.1T + 47T^{2} \) |
| 53 | \( 1 + 7.01T + 53T^{2} \) |
| 59 | \( 1 + 1.74T + 59T^{2} \) |
| 61 | \( 1 - 2.37T + 61T^{2} \) |
| 67 | \( 1 + 0.291T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 12.7T + 73T^{2} \) |
| 79 | \( 1 - 9.95T + 79T^{2} \) |
| 83 | \( 1 - 3.23T + 83T^{2} \) |
| 89 | \( 1 - 8.04T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−8.867581467594802479060751048514, −8.082748921398448298702101028596, −7.75037591718945589825292063401, −6.76038448693690560101050473386, −6.04254588464563659892371638280, −5.08899339013967068552132565931, −3.82190087129718513860113042645, −3.39499799671082661381354628028, −2.02676611753493146340302804626, −0.37412943006194526408055941740,
0.37412943006194526408055941740, 2.02676611753493146340302804626, 3.39499799671082661381354628028, 3.82190087129718513860113042645, 5.08899339013967068552132565931, 6.04254588464563659892371638280, 6.76038448693690560101050473386, 7.75037591718945589825292063401, 8.082748921398448298702101028596, 8.867581467594802479060751048514
