L(s) = 1 | − 0.841·2-s + 0.162·3-s − 1.29·4-s + 0.443·5-s − 0.136·6-s + 0.552·7-s + 2.76·8-s − 2.97·9-s − 0.373·10-s − 3.97·11-s − 0.209·12-s − 13-s − 0.464·14-s + 0.0719·15-s + 0.255·16-s + 2.34·17-s + 2.50·18-s − 2.51·19-s − 0.573·20-s + 0.0895·21-s + 3.34·22-s − 2.52·23-s + 0.448·24-s − 4.80·25-s + 0.841·26-s − 0.968·27-s − 0.713·28-s + ⋯ |
L(s) = 1 | − 0.594·2-s + 0.0935·3-s − 0.646·4-s + 0.198·5-s − 0.0556·6-s + 0.208·7-s + 0.979·8-s − 0.991·9-s − 0.118·10-s − 1.19·11-s − 0.0604·12-s − 0.277·13-s − 0.124·14-s + 0.0185·15-s + 0.0638·16-s + 0.568·17-s + 0.589·18-s − 0.576·19-s − 0.128·20-s + 0.0195·21-s + 0.713·22-s − 0.527·23-s + 0.0916·24-s − 0.960·25-s + 0.164·26-s − 0.186·27-s − 0.134·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + 0.841T + 2T^{2} \) |
| 3 | \( 1 - 0.162T + 3T^{2} \) |
| 5 | \( 1 - 0.443T + 5T^{2} \) |
| 7 | \( 1 - 0.552T + 7T^{2} \) |
| 11 | \( 1 + 3.97T + 11T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 + 8.89T + 29T^{2} \) |
| 37 | \( 1 + 2.01T + 37T^{2} \) |
| 41 | \( 1 + 8.35T + 41T^{2} \) |
| 43 | \( 1 - 1.29T + 43T^{2} \) |
| 47 | \( 1 - 13.1T + 47T^{2} \) |
| 53 | \( 1 + 0.584T + 53T^{2} \) |
| 59 | \( 1 + 3.05T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 1.62T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 6.86T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 - 3.93T + 89T^{2} \) |
| 97 | \( 1 - 8.64T + 97T^{2} \) |
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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
−10.58488590450454826607069667574, −9.870591877577775075289116003930, −8.937920227677876524486438903389, −8.124779807949082121294456163883, −7.47608133504567003213273506569, −5.80494949251687631933455025841, −5.08963879308463555324045935724, −3.69473040411012891935736439119, −2.14198710893907457352447783020, 0,
2.14198710893907457352447783020, 3.69473040411012891935736439119, 5.08963879308463555324045935724, 5.80494949251687631933455025841, 7.47608133504567003213273506569, 8.124779807949082121294456163883, 8.937920227677876524486438903389, 9.870591877577775075289116003930, 10.58488590450454826607069667574