| L(s) = 1 | − 1.29·2-s − 2.20·3-s − 0.331·4-s + 2.25·5-s + 2.84·6-s − 4.62·7-s + 3.01·8-s + 1.85·9-s − 2.91·10-s − 2.87·11-s + 0.730·12-s − 13-s + 5.96·14-s − 4.96·15-s − 3.22·16-s + 6.49·17-s − 2.39·18-s + 8.00·19-s − 0.747·20-s + 10.1·21-s + 3.71·22-s + 23-s − 6.63·24-s + 0.0841·25-s + 1.29·26-s + 2.51·27-s + 1.53·28-s + ⋯ |
| L(s) = 1 | − 0.913·2-s − 1.27·3-s − 0.165·4-s + 1.00·5-s + 1.16·6-s − 1.74·7-s + 1.06·8-s + 0.618·9-s − 0.921·10-s − 0.866·11-s + 0.210·12-s − 0.277·13-s + 1.59·14-s − 1.28·15-s − 0.806·16-s + 1.57·17-s − 0.565·18-s + 1.83·19-s − 0.167·20-s + 2.22·21-s + 0.791·22-s + 0.208·23-s − 1.35·24-s + 0.0168·25-s + 0.253·26-s + 0.484·27-s + 0.289·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4193488516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4193488516\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 1.29T + 2T^{2} \) |
| 3 | \( 1 + 2.20T + 3T^{2} \) |
| 5 | \( 1 - 2.25T + 5T^{2} \) |
| 7 | \( 1 + 4.62T + 7T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 17 | \( 1 - 6.49T + 17T^{2} \) |
| 19 | \( 1 - 8.00T + 19T^{2} \) |
| 29 | \( 1 - 2.37T + 29T^{2} \) |
| 31 | \( 1 - 7.98T + 31T^{2} \) |
| 37 | \( 1 - 1.26T + 37T^{2} \) |
| 41 | \( 1 - 3.98T + 41T^{2} \) |
| 43 | \( 1 + 7.41T + 43T^{2} \) |
| 47 | \( 1 + 2.81T + 47T^{2} \) |
| 53 | \( 1 + 6.19T + 53T^{2} \) |
| 59 | \( 1 - 0.769T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 3.12T + 67T^{2} \) |
| 71 | \( 1 + 2.22T + 71T^{2} \) |
| 73 | \( 1 + 5.21T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 6.74T + 83T^{2} \) |
| 89 | \( 1 - 1.08T + 89T^{2} \) |
| 97 | \( 1 + 5.78T + 97T^{2} \) |
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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
−11.66135253717222644303373995438, −10.22414905917542740350818150345, −10.03597420633824095444523373478, −9.396189672875266720039180098345, −7.900571262143861258399546736989, −6.78224817195907648112416103206, −5.78749336152830361317595051461, −5.09715485836788292702784230465, −3.07387336643146877203963627289, −0.810053266113700717441877273435,
0.810053266113700717441877273435, 3.07387336643146877203963627289, 5.09715485836788292702784230465, 5.78749336152830361317595051461, 6.78224817195907648112416103206, 7.900571262143861258399546736989, 9.396189672875266720039180098345, 10.03597420633824095444523373478, 10.22414905917542740350818150345, 11.66135253717222644303373995438
