L(s) = 1 | + 2-s − 0.642·3-s + 4-s + 5-s − 0.642·6-s + 1.07·7-s + 8-s − 2.58·9-s + 10-s − 6.49·11-s − 0.642·12-s + 13-s + 1.07·14-s − 0.642·15-s + 16-s + 3.21·17-s − 2.58·18-s + 2.94·19-s + 20-s − 0.690·21-s − 6.49·22-s − 6.91·23-s − 0.642·24-s + 25-s + 26-s + 3.58·27-s + 1.07·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.370·3-s + 0.5·4-s + 0.447·5-s − 0.262·6-s + 0.406·7-s + 0.353·8-s − 0.862·9-s + 0.316·10-s − 1.95·11-s − 0.185·12-s + 0.277·13-s + 0.287·14-s − 0.165·15-s + 0.250·16-s + 0.778·17-s − 0.609·18-s + 0.674·19-s + 0.223·20-s − 0.150·21-s − 1.38·22-s − 1.44·23-s − 0.131·24-s + 0.200·25-s + 0.196·26-s + 0.690·27-s + 0.203·28-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 | \( 1 + 0.642T + 3T^{2} \) |
| 7 | \( 1 - 1.07T + 7T^{2} \) |
| 11 | \( 1 + 6.49T + 11T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 19 | \( 1 - 2.94T + 19T^{2} \) |
| 23 | \( 1 + 6.91T + 23T^{2} \) |
| 29 | \( 1 + 8.00T + 29T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + 12.5T + 41T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 - 2.47T + 47T^{2} \) |
| 53 | \( 1 - 2.46T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 1.39T + 61T^{2} \) |
| 67 | \( 1 + 8.07T + 67T^{2} \) |
| 71 | \( 1 + 6.30T + 71T^{2} \) |
| 73 | \( 1 + 7.59T + 73T^{2} \) |
| 79 | \( 1 + 6.27T + 79T^{2} \) |
| 83 | \( 1 + 2.02T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 - 18.3T + 97T^{2} \) |
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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
−7.88505706785551977188943828804, −7.48161600367106291086654384534, −6.24358199903820219619367005977, −5.64635021183159904148624917342, −5.31548890886528691373652036289, −4.46533155092225306713593071262, −3.27860747218273468854131463134, −2.64870957228651598177273917966, −1.63787768964363617144269538442, 0,
1.63787768964363617144269538442, 2.64870957228651598177273917966, 3.27860747218273468854131463134, 4.46533155092225306713593071262, 5.31548890886528691373652036289, 5.64635021183159904148624917342, 6.24358199903820219619367005977, 7.48161600367106291086654384534, 7.88505706785551977188943828804