| L(s) = 1 | + 1.22·2-s − 2.38·3-s − 0.511·4-s − 2.91·6-s + 1.22·7-s − 3.06·8-s + 2.69·9-s + 5.09·11-s + 1.22·12-s + 2.60·13-s + 1.48·14-s − 2.71·16-s + 3.28·18-s + 8.51·19-s − 2.91·21-s + 6.21·22-s − 2.89·23-s + 7.31·24-s + 3.18·26-s + 0.732·27-s − 0.623·28-s + 5.49·29-s + 3.18·31-s + 2.81·32-s − 12.1·33-s − 1.37·36-s + 3.09·37-s + ⋯ |
| L(s) = 1 | + 0.862·2-s − 1.37·3-s − 0.255·4-s − 1.18·6-s + 0.461·7-s − 1.08·8-s + 0.897·9-s + 1.53·11-s + 0.352·12-s + 0.723·13-s + 0.397·14-s − 0.678·16-s + 0.774·18-s + 1.95·19-s − 0.635·21-s + 1.32·22-s − 0.603·23-s + 1.49·24-s + 0.623·26-s + 0.140·27-s − 0.117·28-s + 1.01·29-s + 0.571·31-s + 0.497·32-s − 2.11·33-s − 0.229·36-s + 0.508·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.980703870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.980703870\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 1.22T + 2T^{2} \) |
| 3 | \( 1 + 2.38T + 3T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 - 5.09T + 11T^{2} \) |
| 13 | \( 1 - 2.60T + 13T^{2} \) |
| 19 | \( 1 - 8.51T + 19T^{2} \) |
| 23 | \( 1 + 2.89T + 23T^{2} \) |
| 29 | \( 1 - 5.49T + 29T^{2} \) |
| 31 | \( 1 - 3.18T + 31T^{2} \) |
| 37 | \( 1 - 3.09T + 37T^{2} \) |
| 41 | \( 1 + 0.181T + 41T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 + 8.71T + 47T^{2} \) |
| 53 | \( 1 - 0.167T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 0.293T + 61T^{2} \) |
| 67 | \( 1 + 1.56T + 67T^{2} \) |
| 71 | \( 1 - 6.93T + 71T^{2} \) |
| 73 | \( 1 - 8.04T + 73T^{2} \) |
| 79 | \( 1 + 0.0888T + 79T^{2} \) |
| 83 | \( 1 - 14.8T + 83T^{2} \) |
| 89 | \( 1 + 2.79T + 89T^{2} \) |
| 97 | \( 1 - 3.54T + 97T^{2} \) |
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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
−7.87426307546264987740899511404, −6.73148502149467945532010113634, −6.33969525647206687883409990802, −5.79467782302637750941248046226, −4.95793909633553683690286215767, −4.66419318917752446092193352069, −3.72767532685911194390368713903, −3.11705478731126791805328075115, −1.49949089497666042379444493184, −0.74497121649527636883217597998,
0.74497121649527636883217597998, 1.49949089497666042379444493184, 3.11705478731126791805328075115, 3.72767532685911194390368713903, 4.66419318917752446092193352069, 4.95793909633553683690286215767, 5.79467782302637750941248046226, 6.33969525647206687883409990802, 6.73148502149467945532010113634, 7.87426307546264987740899511404
