| L(s) = 1 | − 2-s + 1.27·3-s + 4-s − 5-s − 1.27·6-s − 0.354·7-s − 8-s − 1.36·9-s + 10-s + 0.305·11-s + 1.27·12-s − 13-s + 0.354·14-s − 1.27·15-s + 16-s − 4.03·17-s + 1.36·18-s − 6.04·19-s − 20-s − 0.452·21-s − 0.305·22-s − 0.505·23-s − 1.27·24-s + 25-s + 26-s − 5.58·27-s − 0.354·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.738·3-s + 0.5·4-s − 0.447·5-s − 0.521·6-s − 0.133·7-s − 0.353·8-s − 0.455·9-s + 0.316·10-s + 0.0922·11-s + 0.369·12-s − 0.277·13-s + 0.0946·14-s − 0.330·15-s + 0.250·16-s − 0.978·17-s + 0.321·18-s − 1.38·19-s − 0.223·20-s − 0.0988·21-s − 0.0652·22-s − 0.105·23-s − 0.260·24-s + 0.200·25-s + 0.196·26-s − 1.07·27-s − 0.0669·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.147513829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.147513829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 - 1.27T + 3T^{2} \) |
| 7 | \( 1 + 0.354T + 7T^{2} \) |
| 11 | \( 1 - 0.305T + 11T^{2} \) |
| 17 | \( 1 + 4.03T + 17T^{2} \) |
| 19 | \( 1 + 6.04T + 19T^{2} \) |
| 23 | \( 1 + 0.505T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 37 | \( 1 - 1.59T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 - 9.23T + 43T^{2} \) |
| 47 | \( 1 - 5.99T + 47T^{2} \) |
| 53 | \( 1 - 2.98T + 53T^{2} \) |
| 59 | \( 1 - 3.68T + 59T^{2} \) |
| 61 | \( 1 - 9.10T + 61T^{2} \) |
| 67 | \( 1 - 6.36T + 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 - 8.47T + 73T^{2} \) |
| 79 | \( 1 - 1.73T + 79T^{2} \) |
| 83 | \( 1 + 12.0T + 83T^{2} \) |
| 89 | \( 1 + 0.654T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.309977824598347263292425079951, −8.080616723845830701307190205520, −7.09359392199222785201235614510, −6.47546282204364159291103388592, −5.63400329256811062976726420812, −4.43851046858665211767038849709, −3.78025224474860266947410154252, −2.62482316396319381466457711423, −2.21413347514570793579769275643, −0.62562905544898734703549908591,
0.62562905544898734703549908591, 2.21413347514570793579769275643, 2.62482316396319381466457711423, 3.78025224474860266947410154252, 4.43851046858665211767038849709, 5.63400329256811062976726420812, 6.47546282204364159291103388592, 7.09359392199222785201235614510, 8.080616723845830701307190205520, 8.309977824598347263292425079951
