| L(s) = 1 | − 4·2-s − 25.9·3-s + 16·4-s − 18.4·5-s + 103.·6-s − 122.·7-s − 64·8-s + 428.·9-s + 73.8·10-s − 71.1·11-s − 414.·12-s − 540.·13-s + 488.·14-s + 478.·15-s + 256·16-s + 221.·17-s − 1.71e3·18-s − 295.·20-s + 3.16e3·21-s + 284.·22-s + 4.11e3·23-s + 1.65e3·24-s − 2.78e3·25-s + 2.16e3·26-s − 4.79e3·27-s − 1.95e3·28-s − 5.55e3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.66·3-s + 0.5·4-s − 0.330·5-s + 1.17·6-s − 0.941·7-s − 0.353·8-s + 1.76·9-s + 0.233·10-s − 0.177·11-s − 0.830·12-s − 0.886·13-s + 0.665·14-s + 0.548·15-s + 0.250·16-s + 0.185·17-s − 1.24·18-s − 0.165·20-s + 1.56·21-s + 0.125·22-s + 1.62·23-s + 0.587·24-s − 0.890·25-s + 0.627·26-s − 1.26·27-s − 0.470·28-s − 1.22·29-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
| \(L(3)\) |
\(\approx\) |
\(0.04842459916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04842459916\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 25.9T + 243T^{2} \) |
| 5 | \( 1 + 18.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 122.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 71.1T + 1.61e5T^{2} \) |
| 13 | \( 1 + 540.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 221.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 4.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 6.11e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.03e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.10e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.25e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.44e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.23e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.94e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.60e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.90e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.67e4T + 8.58e9T^{2} \) |
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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
−9.656041536610397464618405322763, −9.144044179975301906885111611017, −7.48059783451550668366435731972, −7.17967492900776576313317903170, −6.06701524339379539649630368701, −5.47152885985903921796014489928, −4.32548732695687081590615704462, −2.99064781104435781391588030963, −1.43794796342737020082405103729, −0.12948662070455469143263854553,
0.12948662070455469143263854553, 1.43794796342737020082405103729, 2.99064781104435781391588030963, 4.32548732695687081590615704462, 5.47152885985903921796014489928, 6.06701524339379539649630368701, 7.17967492900776576313317903170, 7.48059783451550668366435731972, 9.144044179975301906885111611017, 9.656041536610397464618405322763
