| L(s) = 1 | − 2.66·2-s − 0.492·3-s + 5.12·4-s + 2.37·5-s + 1.31·6-s + 1.32·7-s − 8.34·8-s − 2.75·9-s − 6.33·10-s + 0.0220·11-s − 2.52·12-s − 4.40·13-s − 3.53·14-s − 1.16·15-s + 12.0·16-s + 1.88·17-s + 7.36·18-s + 2.05·19-s + 12.1·20-s − 0.652·21-s − 0.0588·22-s + 3.94·23-s + 4.11·24-s + 0.630·25-s + 11.7·26-s + 2.83·27-s + 6.78·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 0.284·3-s + 2.56·4-s + 1.06·5-s + 0.537·6-s + 0.500·7-s − 2.95·8-s − 0.919·9-s − 2.00·10-s + 0.00664·11-s − 0.729·12-s − 1.22·13-s − 0.944·14-s − 0.301·15-s + 3.00·16-s + 0.456·17-s + 1.73·18-s + 0.470·19-s + 2.72·20-s − 0.142·21-s − 0.0125·22-s + 0.822·23-s + 0.839·24-s + 0.126·25-s + 2.30·26-s + 0.546·27-s + 1.28·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6866092135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6866092135\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4027 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.66T + 2T^{2} \) |
| 3 | \( 1 + 0.492T + 3T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 - 1.32T + 7T^{2} \) |
| 11 | \( 1 - 0.0220T + 11T^{2} \) |
| 13 | \( 1 + 4.40T + 13T^{2} \) |
| 17 | \( 1 - 1.88T + 17T^{2} \) |
| 19 | \( 1 - 2.05T + 19T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 + 8.40T + 29T^{2} \) |
| 31 | \( 1 + 5.95T + 31T^{2} \) |
| 37 | \( 1 + 6.74T + 37T^{2} \) |
| 41 | \( 1 + 1.19T + 41T^{2} \) |
| 43 | \( 1 - 9.27T + 43T^{2} \) |
| 47 | \( 1 - 3.38T + 47T^{2} \) |
| 53 | \( 1 + 8.44T + 53T^{2} \) |
| 59 | \( 1 - 8.17T + 59T^{2} \) |
| 61 | \( 1 - 7.87T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 8.53T + 73T^{2} \) |
| 79 | \( 1 - 2.13T + 79T^{2} \) |
| 83 | \( 1 + 9.19T + 83T^{2} \) |
| 89 | \( 1 - 16.1T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−8.574539186427042629010192999334, −7.79434600246674502870000504366, −7.23230605620854565402776142661, −6.49995163151898806075702158581, −5.52146344084077458464016396957, −5.26572879932869854580006914134, −3.38654641166257953117981238636, −2.35477262891348154163864484627, −1.83766884577287444531807312610, −0.62230047909738313659484024373,
0.62230047909738313659484024373, 1.83766884577287444531807312610, 2.35477262891348154163864484627, 3.38654641166257953117981238636, 5.26572879932869854580006914134, 5.52146344084077458464016396957, 6.49995163151898806075702158581, 7.23230605620854565402776142661, 7.79434600246674502870000504366, 8.574539186427042629010192999334
