| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 73.1·5-s + 36·6-s + 90.1·7-s − 64·8-s + 81·9-s + 292.·10-s + 481.·11-s − 144·12-s − 57.0·13-s − 360.·14-s + 658.·15-s + 256·16-s − 1.08e3·17-s − 324·18-s + 2.24e3·19-s − 1.17e3·20-s − 811.·21-s − 1.92e3·22-s − 529·23-s + 576·24-s + 2.22e3·25-s + 228.·26-s − 729·27-s + 1.44e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.30·5-s + 0.408·6-s + 0.695·7-s − 0.353·8-s + 0.333·9-s + 0.925·10-s + 1.19·11-s − 0.288·12-s − 0.0936·13-s − 0.491·14-s + 0.755·15-s + 0.250·16-s − 0.910·17-s − 0.235·18-s + 1.42·19-s − 0.654·20-s − 0.401·21-s − 0.847·22-s − 0.208·23-s + 0.204·24-s + 0.712·25-s + 0.0662·26-s − 0.192·27-s + 0.347·28-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
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 73.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 90.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 481.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 57.0T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.08e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.24e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 4.47e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.87e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.05e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 4.01e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.87e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.20e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.60e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.60e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.84e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.75e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.46e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.42e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.23e5T + 8.58e9T^{2} \) |
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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
−11.64371376006668782627161241807, −11.00909910840574941484123889916, −9.637202500280959376222003680724, −8.488993135427417842693109275030, −7.53071192333979073061545104742, −6.57094086035408450335056739671, −4.90032766202886901807842011605, −3.62124609818277329604769581681, −1.43204416460074913665357417615, 0,
1.43204416460074913665357417615, 3.62124609818277329604769581681, 4.90032766202886901807842011605, 6.57094086035408450335056739671, 7.53071192333979073061545104742, 8.488993135427417842693109275030, 9.637202500280959376222003680724, 11.00909910840574941484123889916, 11.64371376006668782627161241807
