| L(s) = 1 | − 2.61·2-s + 1.19·3-s + 4.82·4-s + 4.07·5-s − 3.12·6-s + 3.61·7-s − 7.39·8-s − 1.56·9-s − 10.6·10-s − 11-s + 5.77·12-s − 1.47·13-s − 9.45·14-s + 4.87·15-s + 9.66·16-s − 3.27·17-s + 4.10·18-s + 19-s + 19.6·20-s + 4.32·21-s + 2.61·22-s − 7.45·23-s − 8.84·24-s + 11.6·25-s + 3.86·26-s − 5.46·27-s + 17.4·28-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 0.690·3-s + 2.41·4-s + 1.82·5-s − 1.27·6-s + 1.36·7-s − 2.61·8-s − 0.523·9-s − 3.36·10-s − 0.301·11-s + 1.66·12-s − 0.410·13-s − 2.52·14-s + 1.25·15-s + 2.41·16-s − 0.793·17-s + 0.966·18-s + 0.229·19-s + 4.40·20-s + 0.944·21-s + 0.557·22-s − 1.55·23-s − 1.80·24-s + 2.32·25-s + 0.757·26-s − 1.05·27-s + 3.30·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 209 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9090106961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9090106961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 2.61T + 2T^{2} \) |
| 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 7 | \( 1 - 3.61T + 7T^{2} \) |
| 13 | \( 1 + 1.47T + 13T^{2} \) |
| 17 | \( 1 + 3.27T + 17T^{2} \) |
| 23 | \( 1 + 7.45T + 23T^{2} \) |
| 29 | \( 1 - 1.02T + 29T^{2} \) |
| 31 | \( 1 - 1.64T + 31T^{2} \) |
| 37 | \( 1 + 6.71T + 37T^{2} \) |
| 41 | \( 1 + 3.92T + 41T^{2} \) |
| 43 | \( 1 - 5.38T + 43T^{2} \) |
| 47 | \( 1 + 3.71T + 47T^{2} \) |
| 53 | \( 1 + 0.102T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 + 6.49T + 61T^{2} \) |
| 67 | \( 1 + 3.70T + 67T^{2} \) |
| 71 | \( 1 - 6.32T + 71T^{2} \) |
| 73 | \( 1 + 1.37T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 5.44T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−11.97346819375939427866831060834, −10.91777465262375145779190572583, −10.15033492401553141639253930737, −9.326062409880506864726372491087, −8.565913160388211801843917993779, −7.83985106599742827476942966363, −6.52130790402011248370966936784, −5.35883619559174249974714204608, −2.44642084002183667295276153334, −1.79382865408832722659243733553,
1.79382865408832722659243733553, 2.44642084002183667295276153334, 5.35883619559174249974714204608, 6.52130790402011248370966936784, 7.83985106599742827476942966363, 8.565913160388211801843917993779, 9.326062409880506864726372491087, 10.15033492401553141639253930737, 10.91777465262375145779190572583, 11.97346819375939427866831060834
