| L(s) = 1 | + 2-s − 3-s + 4-s + 1.37·5-s − 6-s + 4.23·7-s + 8-s + 9-s + 1.37·10-s + 11-s − 12-s − 5.14·13-s + 4.23·14-s − 1.37·15-s + 16-s − 4.20·17-s + 18-s + 7.86·19-s + 1.37·20-s − 4.23·21-s + 22-s + 4.25·23-s − 24-s − 3.11·25-s − 5.14·26-s − 27-s + 4.23·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.613·5-s − 0.408·6-s + 1.60·7-s + 0.353·8-s + 0.333·9-s + 0.433·10-s + 0.301·11-s − 0.288·12-s − 1.42·13-s + 1.13·14-s − 0.354·15-s + 0.250·16-s − 1.01·17-s + 0.235·18-s + 1.80·19-s + 0.306·20-s − 0.923·21-s + 0.213·22-s + 0.887·23-s − 0.204·24-s − 0.623·25-s − 1.00·26-s − 0.192·27-s + 0.800·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.529219430\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.529219430\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 5 | \( 1 - 1.37T + 5T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 13 | \( 1 + 5.14T + 13T^{2} \) |
| 17 | \( 1 + 4.20T + 17T^{2} \) |
| 19 | \( 1 - 7.86T + 19T^{2} \) |
| 23 | \( 1 - 4.25T + 23T^{2} \) |
| 29 | \( 1 - 1.03T + 29T^{2} \) |
| 31 | \( 1 - 4.77T + 31T^{2} \) |
| 37 | \( 1 + 1.57T + 37T^{2} \) |
| 41 | \( 1 - 10.4T + 41T^{2} \) |
| 43 | \( 1 + 1.91T + 43T^{2} \) |
| 47 | \( 1 - 5.37T + 47T^{2} \) |
| 53 | \( 1 + 6.73T + 53T^{2} \) |
| 59 | \( 1 - 6.26T + 59T^{2} \) |
| 67 | \( 1 + 1.68T + 67T^{2} \) |
| 71 | \( 1 - 6.54T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 3.88T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 6.47T + 89T^{2} \) |
| 97 | \( 1 - 3.40T + 97T^{2} \) |
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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
−8.286069428092319947778335376149, −7.43051584727201002060319015744, −7.04873984829850724281772750314, −5.98539378297663985765952366392, −5.31305612197261570890172625621, −4.81136077351872981491203586629, −4.22467780332994002063946383692, −2.83411440003332910370433401340, −2.02077643168931595756146249273, −1.07137586310736095381193208687,
1.07137586310736095381193208687, 2.02077643168931595756146249273, 2.83411440003332910370433401340, 4.22467780332994002063946383692, 4.81136077351872981491203586629, 5.31305612197261570890172625621, 5.98539378297663985765952366392, 7.04873984829850724281772750314, 7.43051584727201002060319015744, 8.286069428092319947778335376149
