| L(s) = 1 | − 2.10·2-s − 1.73·3-s + 2.44·4-s + 1.13·5-s + 3.65·6-s + 3.68·7-s − 0.942·8-s − 0.00197·9-s − 2.38·10-s − 11-s − 4.23·12-s + 2.41·13-s − 7.76·14-s − 1.96·15-s − 2.90·16-s + 7.02·17-s + 0.00415·18-s + 6.50·19-s + 2.77·20-s − 6.37·21-s + 2.10·22-s + 7.04·23-s + 1.63·24-s − 3.71·25-s − 5.09·26-s + 5.19·27-s + 9.01·28-s + ⋯ |
| L(s) = 1 | − 1.49·2-s − 0.999·3-s + 1.22·4-s + 0.506·5-s + 1.49·6-s + 1.39·7-s − 0.333·8-s − 0.000656·9-s − 0.754·10-s − 0.301·11-s − 1.22·12-s + 0.669·13-s − 2.07·14-s − 0.506·15-s − 0.726·16-s + 1.70·17-s + 0.000979·18-s + 1.49·19-s + 0.619·20-s − 1.39·21-s + 0.449·22-s + 1.46·23-s + 0.333·24-s − 0.743·25-s − 0.998·26-s + 1.00·27-s + 1.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7884927559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7884927559\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 2.10T + 2T^{2} \) |
| 3 | \( 1 + 1.73T + 3T^{2} \) |
| 5 | \( 1 - 1.13T + 5T^{2} \) |
| 7 | \( 1 - 3.68T + 7T^{2} \) |
| 13 | \( 1 - 2.41T + 13T^{2} \) |
| 17 | \( 1 - 7.02T + 17T^{2} \) |
| 19 | \( 1 - 6.50T + 19T^{2} \) |
| 23 | \( 1 - 7.04T + 23T^{2} \) |
| 29 | \( 1 - 9.17T + 29T^{2} \) |
| 31 | \( 1 + 8.98T + 31T^{2} \) |
| 37 | \( 1 - 0.492T + 37T^{2} \) |
| 41 | \( 1 + 4.27T + 41T^{2} \) |
| 43 | \( 1 + 9.07T + 43T^{2} \) |
| 47 | \( 1 + 1.94T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 9.76T + 59T^{2} \) |
| 61 | \( 1 + 1.61T + 61T^{2} \) |
| 67 | \( 1 + 1.51T + 67T^{2} \) |
| 71 | \( 1 - 5.08T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 - 7.55T + 79T^{2} \) |
| 83 | \( 1 - 8.41T + 83T^{2} \) |
| 89 | \( 1 - 7.35T + 89T^{2} \) |
| 97 | \( 1 - 9.82T + 97T^{2} \) |
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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
−9.595883977655219045117375343329, −8.703467857424754888548791683602, −8.007537392466910134895714722410, −7.39312077005588337900816018065, −6.38499657045742258864368307474, −5.30343916545302719134913514468, −5.01290785599639598018372383235, −3.18222031591274252710782424568, −1.61601950724485278170002034035, −0.938457737900371090719919542274,
0.938457737900371090719919542274, 1.61601950724485278170002034035, 3.18222031591274252710782424568, 5.01290785599639598018372383235, 5.30343916545302719134913514468, 6.38499657045742258864368307474, 7.39312077005588337900816018065, 8.007537392466910134895714722410, 8.703467857424754888548791683602, 9.595883977655219045117375343329
