L(s) = 1 | + 3.24·3-s + 0.246·5-s + 7.54·9-s + 5.04·11-s − 2.10·13-s + 0.801·15-s + 4.96·17-s + 0.506·19-s + 4.93·23-s − 4.93·25-s + 14.7·27-s − 4.66·29-s + 9.07·31-s + 16.3·33-s − 2.64·37-s − 6.85·39-s − 6.70·41-s − 3.62·43-s + 1.86·45-s − 0.917·47-s + 16.1·51-s − 3.62·53-s + 1.24·55-s + 1.64·57-s − 7.67·59-s − 13.0·61-s − 0.521·65-s + ⋯ |
L(s) = 1 | + 1.87·3-s + 0.110·5-s + 2.51·9-s + 1.52·11-s − 0.585·13-s + 0.207·15-s + 1.20·17-s + 0.116·19-s + 1.02·23-s − 0.987·25-s + 2.83·27-s − 0.866·29-s + 1.63·31-s + 2.85·33-s − 0.434·37-s − 1.09·39-s − 1.04·41-s − 0.552·43-s + 0.277·45-s − 0.133·47-s + 2.25·51-s − 0.498·53-s + 0.168·55-s + 0.217·57-s − 0.999·59-s − 1.66·61-s − 0.0646·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.966005975\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.966005975\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 3.24T + 3T^{2} \) |
| 5 | \( 1 - 0.246T + 5T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 13 | \( 1 + 2.10T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 - 0.506T + 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 + 4.66T + 29T^{2} \) |
| 31 | \( 1 - 9.07T + 31T^{2} \) |
| 37 | \( 1 + 2.64T + 37T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 43 | \( 1 + 3.62T + 43T^{2} \) |
| 47 | \( 1 + 0.917T + 47T^{2} \) |
| 53 | \( 1 + 3.62T + 53T^{2} \) |
| 59 | \( 1 + 7.67T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 5.46T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 5.36T + 73T^{2} \) |
| 79 | \( 1 - 6.56T + 79T^{2} \) |
| 83 | \( 1 + 0.753T + 83T^{2} \) |
| 89 | \( 1 - 9.69T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−8.129668307402493564576735327447, −7.64498386347907562586451791957, −6.94283223961599906962076699415, −6.24816975407993068868045199494, −5.03145875352564813677968435985, −4.24460459945231896653069518383, −3.45602347906652399810970253419, −3.01039912798724030417799270949, −1.90814945941182992479148912638, −1.24648246438121246631188848933,
1.24648246438121246631188848933, 1.90814945941182992479148912638, 3.01039912798724030417799270949, 3.45602347906652399810970253419, 4.24460459945231896653069518383, 5.03145875352564813677968435985, 6.24816975407993068868045199494, 6.94283223961599906962076699415, 7.64498386347907562586451791957, 8.129668307402493564576735327447