L(s) = 1 | + 2-s + 2.50·3-s + 4-s − 2.28·5-s + 2.50·6-s − 2.44·7-s + 8-s + 3.28·9-s − 2.28·10-s + 1.22·11-s + 2.50·12-s − 2.44·14-s − 5.72·15-s + 16-s − 4.50·17-s + 3.28·18-s + 19-s − 2.28·20-s − 6.12·21-s + 1.22·22-s − 3.34·23-s + 2.50·24-s + 0.221·25-s + 0.714·27-s − 2.44·28-s − 4.23·29-s − 5.72·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.44·3-s + 0.5·4-s − 1.02·5-s + 1.02·6-s − 0.923·7-s + 0.353·8-s + 1.09·9-s − 0.722·10-s + 0.368·11-s + 0.723·12-s − 0.653·14-s − 1.47·15-s + 0.250·16-s − 1.09·17-s + 0.774·18-s + 0.229·19-s − 0.510·20-s − 1.33·21-s + 0.260·22-s − 0.698·23-s + 0.511·24-s + 0.0443·25-s + 0.137·27-s − 0.461·28-s − 0.786·29-s − 1.04·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2.50T + 3T^{2} \) |
| 5 | \( 1 + 2.28T + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 11 | \( 1 - 1.22T + 11T^{2} \) |
| 17 | \( 1 + 4.50T + 17T^{2} \) |
| 23 | \( 1 + 3.34T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 - 1.29T + 31T^{2} \) |
| 37 | \( 1 + 0.985T + 37T^{2} \) |
| 41 | \( 1 + 1.93T + 41T^{2} \) |
| 43 | \( 1 + 4.44T + 43T^{2} \) |
| 47 | \( 1 - 0.443T + 47T^{2} \) |
| 53 | \( 1 - 8.72T + 53T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 - 0.443T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 4.41T + 71T^{2} \) |
| 73 | \( 1 + 8.69T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 4.18T + 83T^{2} \) |
| 89 | \( 1 + 1.22T + 89T^{2} \) |
| 97 | \( 1 + 11.6T + 97T^{2} \) |
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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
−7.74551722111415233273975606075, −6.98671870049384665500676640945, −6.45607723605422751320688226253, −5.47835814237911057496336680100, −4.32872704406942227867367082558, −3.91971149144226505136700000124, −3.30323563132209955618975472427, −2.63209195700557535955315092278, −1.72490895421668432917715592684, 0,
1.72490895421668432917715592684, 2.63209195700557535955315092278, 3.30323563132209955618975472427, 3.91971149144226505136700000124, 4.32872704406942227867367082558, 5.47835814237911057496336680100, 6.45607723605422751320688226253, 6.98671870049384665500676640945, 7.74551722111415233273975606075