| L(s) = 1 | − 5.05·2-s − 3·3-s + 17.5·4-s − 10.1·5-s + 15.1·6-s − 15.0·7-s − 48.1·8-s + 9·9-s + 51.1·10-s + 11·11-s − 52.5·12-s + 12.9·13-s + 76.1·14-s + 30.3·15-s + 102.·16-s + 42.4·17-s − 45.4·18-s + 46.8·19-s − 177.·20-s + 45.1·21-s − 55.5·22-s + 130.·23-s + 144.·24-s − 22.5·25-s − 65.4·26-s − 27·27-s − 263.·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 0.577·3-s + 2.19·4-s − 0.905·5-s + 1.03·6-s − 0.813·7-s − 2.12·8-s + 0.333·9-s + 1.61·10-s + 0.301·11-s − 1.26·12-s + 0.276·13-s + 1.45·14-s + 0.522·15-s + 1.60·16-s + 0.606·17-s − 0.595·18-s + 0.566·19-s − 1.98·20-s + 0.469·21-s − 0.538·22-s + 1.17·23-s + 1.22·24-s − 0.180·25-s − 0.494·26-s − 0.192·27-s − 1.78·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5552791979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5552791979\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 5.05T + 8T^{2} \) |
| 5 | \( 1 + 10.1T + 125T^{2} \) |
| 7 | \( 1 + 15.0T + 343T^{2} \) |
| 13 | \( 1 - 12.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 42.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 46.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 130.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 37.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 334.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 319.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 216.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 382.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 334.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 517.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 828.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 88.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 59.9T + 3.57e5T^{2} \) |
| 73 | \( 1 - 456.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 912.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 549.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 28.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 751.T + 9.12e5T^{2} \) |
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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
−8.878324472570030276320967870351, −7.957910268964433553522479964286, −7.52729013151312596695644505776, −6.60474519639771443839855876883, −6.15025633545714167517259154178, −4.80554989873738541291123666731, −3.57243485605481925915690410074, −2.67714775362053645756974672663, −1.16247879997165998322093161431, −0.56032745250932676189353434382,
0.56032745250932676189353434382, 1.16247879997165998322093161431, 2.67714775362053645756974672663, 3.57243485605481925915690410074, 4.80554989873738541291123666731, 6.15025633545714167517259154178, 6.60474519639771443839855876883, 7.52729013151312596695644505776, 7.957910268964433553522479964286, 8.878324472570030276320967870351
