| L(s) = 1 | + 4·2-s + 17.4·3-s + 16·4-s − 79.4·5-s + 69.9·6-s + 132.·7-s + 64·8-s + 62.5·9-s − 317.·10-s + 311.·11-s + 279.·12-s − 901.·13-s + 531.·14-s − 1.38e3·15-s + 256·16-s − 157.·17-s + 250.·18-s − 1.27e3·20-s + 2.32e3·21-s + 1.24e3·22-s − 2.52e3·23-s + 1.11e3·24-s + 3.18e3·25-s − 3.60e3·26-s − 3.15e3·27-s + 2.12e3·28-s − 4.73e3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.12·3-s + 0.5·4-s − 1.42·5-s + 0.792·6-s + 1.02·7-s + 0.353·8-s + 0.257·9-s − 1.00·10-s + 0.776·11-s + 0.560·12-s − 1.47·13-s + 0.724·14-s − 1.59·15-s + 0.250·16-s − 0.132·17-s + 0.182·18-s − 0.710·20-s + 1.14·21-s + 0.548·22-s − 0.994·23-s + 0.396·24-s + 1.01·25-s − 1.04·26-s − 0.832·27-s + 0.512·28-s − 1.04·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 17.4T + 243T^{2} \) |
| 5 | \( 1 + 79.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 132.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 311.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 901.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 157.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.52e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.97e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.09e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.50e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.16e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 6.15e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.27e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.53e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.10e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.10e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.70e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.39e5T + 8.58e9T^{2} \) |
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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
−8.972836679031130040945183979554, −8.110705677802465804289182662283, −7.67154200827645059263701850374, −6.83437592666392758265610783377, −5.31116791152855112130258181035, −4.34445457776065799719279065599, −3.76051539946298127985799676609, −2.70763265176811655994264136006, −1.70034429193977377944645637006, 0,
1.70034429193977377944645637006, 2.70763265176811655994264136006, 3.76051539946298127985799676609, 4.34445457776065799719279065599, 5.31116791152855112130258181035, 6.83437592666392758265610783377, 7.67154200827645059263701850374, 8.110705677802465804289182662283, 8.972836679031130040945183979554
