| L(s) = 1 | + 2-s − 2.66·3-s + 4-s + 5-s − 2.66·6-s − 4.14·7-s + 8-s + 4.09·9-s + 10-s + 1.20·11-s − 2.66·12-s − 13-s − 4.14·14-s − 2.66·15-s + 16-s − 2.37·17-s + 4.09·18-s + 7.96·19-s + 20-s + 11.0·21-s + 1.20·22-s − 3.02·23-s − 2.66·24-s + 25-s − 26-s − 2.90·27-s − 4.14·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.53·3-s + 0.5·4-s + 0.447·5-s − 1.08·6-s − 1.56·7-s + 0.353·8-s + 1.36·9-s + 0.316·10-s + 0.363·11-s − 0.768·12-s − 0.277·13-s − 1.10·14-s − 0.687·15-s + 0.250·16-s − 0.575·17-s + 0.964·18-s + 1.82·19-s + 0.223·20-s + 2.40·21-s + 0.256·22-s − 0.631·23-s − 0.543·24-s + 0.200·25-s − 0.196·26-s − 0.559·27-s − 0.782·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 3 | \( 1 + 2.66T + 3T^{2} \) |
| 7 | \( 1 + 4.14T + 7T^{2} \) |
| 11 | \( 1 - 1.20T + 11T^{2} \) |
| 17 | \( 1 + 2.37T + 17T^{2} \) |
| 19 | \( 1 - 7.96T + 19T^{2} \) |
| 23 | \( 1 + 3.02T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 37 | \( 1 - 6.34T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + 5.81T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 4.60T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 12.8T + 73T^{2} \) |
| 79 | \( 1 + 2.00T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 - 3.01T + 89T^{2} \) |
| 97 | \( 1 + 2.69T + 97T^{2} \) |
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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
−7.64565463244826001490500573628, −7.00134206325677228590741563179, −6.27270552463773342440705622106, −5.95742161257375630467038954663, −5.26194340618649135722226722660, −4.44069473784300727360716472616, −3.52275336000762671731992147010, −2.65442331886384740805571900880, −1.25945716079953330796388844086, 0,
1.25945716079953330796388844086, 2.65442331886384740805571900880, 3.52275336000762671731992147010, 4.44069473784300727360716472616, 5.26194340618649135722226722660, 5.95742161257375630467038954663, 6.27270552463773342440705622106, 7.00134206325677228590741563179, 7.64565463244826001490500573628
