| L(s) = 1 | + 0.0765·2-s − 3-s − 1.99·4-s + 2.75·5-s − 0.0765·6-s − 0.305·8-s + 9-s + 0.210·10-s + 4.59·11-s + 1.99·12-s + 4.62·13-s − 2.75·15-s + 3.96·16-s − 5.10·17-s + 0.0765·18-s + 5.16·19-s − 5.48·20-s + 0.351·22-s + 23-s + 0.305·24-s + 2.57·25-s + 0.354·26-s − 27-s + 0.812·29-s − 0.210·30-s + 4.06·31-s + 0.914·32-s + ⋯ |
| L(s) = 1 | + 0.0541·2-s − 0.577·3-s − 0.997·4-s + 1.23·5-s − 0.0312·6-s − 0.108·8-s + 0.333·9-s + 0.0666·10-s + 1.38·11-s + 0.575·12-s + 1.28·13-s − 0.710·15-s + 0.991·16-s − 1.23·17-s + 0.0180·18-s + 1.18·19-s − 1.22·20-s + 0.0749·22-s + 0.208·23-s + 0.0624·24-s + 0.515·25-s + 0.0695·26-s − 0.192·27-s + 0.150·29-s − 0.0384·30-s + 0.730·31-s + 0.161·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.891477761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.891477761\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0765T + 2T^{2} \) |
| 5 | \( 1 - 2.75T + 5T^{2} \) |
| 11 | \( 1 - 4.59T + 11T^{2} \) |
| 13 | \( 1 - 4.62T + 13T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 - 5.16T + 19T^{2} \) |
| 29 | \( 1 - 0.812T + 29T^{2} \) |
| 31 | \( 1 - 4.06T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 + 2.39T + 41T^{2} \) |
| 43 | \( 1 + 11.9T + 43T^{2} \) |
| 47 | \( 1 + 7.22T + 47T^{2} \) |
| 53 | \( 1 - 3.67T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 5.09T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 4.57T + 71T^{2} \) |
| 73 | \( 1 + 10.2T + 73T^{2} \) |
| 79 | \( 1 - 5.11T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 7.06T + 89T^{2} \) |
| 97 | \( 1 - 0.960T + 97T^{2} \) |
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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
−8.868428726633114172962646346738, −8.045681546853726589330250610357, −6.76320718008115737100111801962, −6.28246422705226633137101646191, −5.66484882613474846257456216255, −4.81611085102224735866988022787, −4.11067873187726467540030928879, −3.18463719028324624161609726202, −1.70078080703227541456886316469, −0.927937318403193030787249752430,
0.927937318403193030787249752430, 1.70078080703227541456886316469, 3.18463719028324624161609726202, 4.11067873187726467540030928879, 4.81611085102224735866988022787, 5.66484882613474846257456216255, 6.28246422705226633137101646191, 6.76320718008115737100111801962, 8.045681546853726589330250610357, 8.868428726633114172962646346738
