| L(s) = 1 | − 4.32·2-s + 10.7·4-s − 4.62·5-s + 32.0·7-s − 11.8·8-s + 20.0·10-s + 7.45·11-s + 58.3·13-s − 138.·14-s − 34.5·16-s − 66.5·17-s − 121.·19-s − 49.6·20-s − 32.2·22-s − 161.·23-s − 103.·25-s − 252.·26-s + 343.·28-s + 33.6·29-s − 86.2·31-s + 244.·32-s + 288.·34-s − 148.·35-s + 122.·37-s + 528.·38-s + 54.7·40-s − 158.·41-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + 1.34·4-s − 0.413·5-s + 1.72·7-s − 0.523·8-s + 0.632·10-s + 0.204·11-s + 1.24·13-s − 2.64·14-s − 0.540·16-s − 0.949·17-s − 1.47·19-s − 0.554·20-s − 0.312·22-s − 1.46·23-s − 0.829·25-s − 1.90·26-s + 2.32·28-s + 0.215·29-s − 0.499·31-s + 1.35·32-s + 1.45·34-s − 0.714·35-s + 0.546·37-s + 2.25·38-s + 0.216·40-s − 0.603·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9382185980\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9382185980\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 + 4.32T + 8T^{2} \) |
| 5 | \( 1 + 4.62T + 125T^{2} \) |
| 7 | \( 1 - 32.0T + 343T^{2} \) |
| 11 | \( 1 - 7.45T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 66.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 121.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 161.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 33.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 86.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 122.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 158.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 429.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 400.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 663.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 609.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 425.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 473.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 120.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 120.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 830.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 188.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.69e3T + 9.12e5T^{2} \) |
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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
−8.521888853112878249125772016309, −8.220407876930989122565103754230, −7.60039717936210497206554684398, −6.63912330655830353020648264517, −5.81071295613954832473830459684, −4.47546949768898609331486578037, −4.00686356160437294935826336356, −2.16059879271031816291856873204, −1.69749968715420452008353070017, −0.56745291664274414834599674240,
0.56745291664274414834599674240, 1.69749968715420452008353070017, 2.16059879271031816291856873204, 4.00686356160437294935826336356, 4.47546949768898609331486578037, 5.81071295613954832473830459684, 6.63912330655830353020648264517, 7.60039717936210497206554684398, 8.220407876930989122565103754230, 8.521888853112878249125772016309
