| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s − 33.6·5-s + 36·6-s − 116.·7-s − 64·8-s + 81·9-s + 134.·10-s − 92.6·11-s − 144·12-s − 695.·13-s + 466.·14-s + 303.·15-s + 256·16-s − 810.·17-s − 324·18-s + 256.·19-s − 538.·20-s + 1.05e3·21-s + 370.·22-s − 4.49e3·23-s + 576·24-s − 1.99e3·25-s + 2.78e3·26-s − 729·27-s − 1.86e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.602·5-s + 0.408·6-s − 0.900·7-s − 0.353·8-s + 0.333·9-s + 0.425·10-s − 0.230·11-s − 0.288·12-s − 1.14·13-s + 0.636·14-s + 0.347·15-s + 0.250·16-s − 0.679·17-s − 0.235·18-s + 0.162·19-s − 0.301·20-s + 0.519·21-s + 0.163·22-s − 1.77·23-s + 0.204·24-s − 0.637·25-s + 0.806·26-s − 0.192·27-s − 0.450·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1574969708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1574969708\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 9T \) |
| 59 | \( 1 - 3.48e3T \) |
| good | 5 | \( 1 + 33.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 116.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 92.6T + 1.61e5T^{2} \) |
| 13 | \( 1 + 695.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 810.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 256.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.49e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.54e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.01e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.98e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.02e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.99e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.15e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 1.60e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.87e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.18e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.46e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.28e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.04e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.19e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.64e4T + 8.58e9T^{2} \) |
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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
−10.31893030408424537227341954371, −9.993916098357047493548570493760, −8.815006320827174708503885671997, −7.76541996971561974040027571480, −6.92390614351895058168304541263, −6.03188676522037713337325037211, −4.72487596350321619461162690342, −3.42035141572560232761162566658, −2.03084294299612777385365779972, −0.23179998803958650912217498006,
0.23179998803958650912217498006, 2.03084294299612777385365779972, 3.42035141572560232761162566658, 4.72487596350321619461162690342, 6.03188676522037713337325037211, 6.92390614351895058168304541263, 7.76541996971561974040027571480, 8.815006320827174708503885671997, 9.993916098357047493548570493760, 10.31893030408424537227341954371
