L(s) = 1 | + 0.713·2-s − 3.20·3-s − 1.49·4-s + 2·5-s − 2.28·6-s + 7-s − 2.49·8-s + 7.26·9-s + 1.42·10-s − 2·11-s + 4.77·12-s + 5.49·13-s + 0.713·14-s − 6.40·15-s + 1.20·16-s + 5.69·17-s + 5.18·18-s + 4.71·19-s − 2.98·20-s − 3.20·21-s − 1.42·22-s − 0.795·23-s + 7.98·24-s − 25-s + 3.91·26-s − 13.6·27-s − 1.49·28-s + ⋯ |
L(s) = 1 | + 0.504·2-s − 1.85·3-s − 0.745·4-s + 0.894·5-s − 0.933·6-s + 0.377·7-s − 0.880·8-s + 2.42·9-s + 0.451·10-s − 0.603·11-s + 1.37·12-s + 1.52·13-s + 0.190·14-s − 1.65·15-s + 0.301·16-s + 1.38·17-s + 1.22·18-s + 1.08·19-s − 0.666·20-s − 0.699·21-s − 0.304·22-s − 0.165·23-s + 1.62·24-s − 0.200·25-s + 0.768·26-s − 2.63·27-s − 0.281·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9550557511\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9550557511\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
good | 2 | \( 1 - 0.713T + 2T^{2} \) |
| 3 | \( 1 + 3.20T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 5.49T + 13T^{2} \) |
| 17 | \( 1 - 5.69T + 17T^{2} \) |
| 19 | \( 1 - 4.71T + 19T^{2} \) |
| 23 | \( 1 + 0.795T + 23T^{2} \) |
| 29 | \( 1 - 2.57T + 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 43 | \( 1 - 11.8T + 43T^{2} \) |
| 47 | \( 1 - 5.89T + 47T^{2} \) |
| 53 | \( 1 + 3.55T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 5.55T + 61T^{2} \) |
| 67 | \( 1 - 7.55T + 67T^{2} \) |
| 71 | \( 1 - 1.83T + 71T^{2} \) |
| 73 | \( 1 - 9.39T + 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 - 0.854T + 83T^{2} \) |
| 89 | \( 1 - 7.19T + 89T^{2} \) |
| 97 | \( 1 - 5.36T + 97T^{2} \) |
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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
−12.00188303282130459037842533983, −10.88906846020601049703707396994, −10.20987026111472155959273284815, −9.262319379464032841659332933605, −7.78715544269595362436534009089, −6.31791268785482116635448179727, −5.55491939461407932527332862018, −5.15769851138017662170288207288, −3.75202276092539216802942047422, −1.13914682079431735717257681685,
1.13914682079431735717257681685, 3.75202276092539216802942047422, 5.15769851138017662170288207288, 5.55491939461407932527332862018, 6.31791268785482116635448179727, 7.78715544269595362436534009089, 9.262319379464032841659332933605, 10.20987026111472155959273284815, 10.88906846020601049703707396994, 12.00188303282130459037842533983