| L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 48.0·5-s − 36·6-s − 163.·7-s + 64·8-s + 81·9-s − 192.·10-s + 261.·11-s − 144·12-s + 860.·13-s − 653.·14-s + 432.·15-s + 256·16-s + 676.·17-s + 324·18-s + 1.93e3·19-s − 768.·20-s + 1.47e3·21-s + 1.04e3·22-s − 529·23-s − 576·24-s − 819.·25-s + 3.44e3·26-s − 729·27-s − 2.61e3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.858·5-s − 0.408·6-s − 1.26·7-s + 0.353·8-s + 0.333·9-s − 0.607·10-s + 0.651·11-s − 0.288·12-s + 1.41·13-s − 0.891·14-s + 0.495·15-s + 0.250·16-s + 0.567·17-s + 0.235·18-s + 1.23·19-s − 0.429·20-s + 0.727·21-s + 0.460·22-s − 0.208·23-s − 0.204·24-s − 0.262·25-s + 0.998·26-s − 0.192·27-s − 0.630·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.980088879\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.980088879\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 + 9T \) |
| 23 | \( 1 + 529T \) |
| good | 5 | \( 1 + 48.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 163.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 261.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 860.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 676.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.93e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 4.96e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.98e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.17e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 6.60e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.84e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.81e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.78e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.66e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.32e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.38e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.73e5T + 8.58e9T^{2} \) |
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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
−12.16985393716468678787290119747, −11.57121869580930344449111704392, −10.46074295268657719538521051092, −9.247989103840756074716273785428, −7.71955263469418789642141080790, −6.56993007150118389262693954562, −5.73201844459514051818983734533, −4.09394241689977808636657504783, −3.26913690441940758785273466900, −0.902596065726272207102489256856,
0.902596065726272207102489256856, 3.26913690441940758785273466900, 4.09394241689977808636657504783, 5.73201844459514051818983734533, 6.56993007150118389262693954562, 7.71955263469418789642141080790, 9.247989103840756074716273785428, 10.46074295268657719538521051092, 11.57121869580930344449111704392, 12.16985393716468678787290119747
