| L(s) = 1 | + 4·2-s − 19.8·3-s + 16·4-s + 105.·5-s − 79.4·6-s + 36.4·7-s + 64·8-s + 151.·9-s + 422.·10-s + 514.·11-s − 317.·12-s − 314.·13-s + 145.·14-s − 2.09e3·15-s + 256·16-s + 2.28e3·17-s + 607.·18-s + 1.69e3·20-s − 724.·21-s + 2.05e3·22-s − 2.01e3·23-s − 1.27e3·24-s + 8.03e3·25-s − 1.25e3·26-s + 1.80e3·27-s + 583.·28-s − 1.11e3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.27·3-s + 0.5·4-s + 1.88·5-s − 0.901·6-s + 0.281·7-s + 0.353·8-s + 0.625·9-s + 1.33·10-s + 1.28·11-s − 0.637·12-s − 0.515·13-s + 0.198·14-s − 2.40·15-s + 0.250·16-s + 1.91·17-s + 0.442·18-s + 0.944·20-s − 0.358·21-s + 0.906·22-s − 0.795·23-s − 0.450·24-s + 2.57·25-s − 0.364·26-s + 0.477·27-s + 0.140·28-s − 0.246·29-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
| \(L(3)\) |
\(\approx\) |
\(4.189138335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.189138335\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 19.8T + 243T^{2} \) |
| 5 | \( 1 - 105.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 36.4T + 1.68e4T^{2} \) |
| 11 | \( 1 - 514.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 314.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.28e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 2.01e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.17e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.29e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.12e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.79e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.62e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.06e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.44e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.01e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.09e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.25e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.30e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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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
−9.979603882507796539630558088365, −9.082554589960137960205374028015, −7.58857722035426645854643786685, −6.35403661247059444950449409742, −6.09367929108362841648674298513, −5.33611673158855064838751836799, −4.56629898520724974922981310925, −3.05887290498188058536388123384, −1.74821298558402429281966932463, −1.00639893515861149237561115790,
1.00639893515861149237561115790, 1.74821298558402429281966932463, 3.05887290498188058536388123384, 4.56629898520724974922981310925, 5.33611673158855064838751836799, 6.09367929108362841648674298513, 6.35403661247059444950449409742, 7.58857722035426645854643786685, 9.082554589960137960205374028015, 9.979603882507796539630558088365
