| L(s) = 1 | + 7.56·2-s − 9·3-s + 25.2·4-s + 40.1·5-s − 68.0·6-s + 194.·7-s − 51.1·8-s + 81·9-s + 303.·10-s − 39.1·11-s − 227.·12-s + 705.·13-s + 1.47e3·14-s − 361.·15-s − 1.19e3·16-s + 1.22e3·17-s + 612.·18-s + 1.89e3·19-s + 1.01e3·20-s − 1.75e3·21-s − 296.·22-s + 529·23-s + 460.·24-s − 1.51e3·25-s + 5.33e3·26-s − 729·27-s + 4.91e3·28-s + ⋯ |
| L(s) = 1 | + 1.33·2-s − 0.577·3-s + 0.788·4-s + 0.718·5-s − 0.772·6-s + 1.50·7-s − 0.282·8-s + 0.333·9-s + 0.960·10-s − 0.0976·11-s − 0.455·12-s + 1.15·13-s + 2.01·14-s − 0.414·15-s − 1.16·16-s + 1.02·17-s + 0.445·18-s + 1.20·19-s + 0.566·20-s − 0.868·21-s − 0.130·22-s + 0.208·23-s + 0.163·24-s − 0.484·25-s + 1.54·26-s − 0.192·27-s + 1.18·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.559958055\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.559958055\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 7.56T + 32T^{2} \) |
| 5 | \( 1 - 40.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 194.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 39.1T + 1.61e5T^{2} \) |
| 13 | \( 1 - 705.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.22e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.89e3T + 2.47e6T^{2} \) |
| 29 | \( 1 + 8.57e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.14e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 4.83e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 222.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.57e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.69e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.02e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.06e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.56e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.30e4T + 8.58e9T^{2} \) |
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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
−13.83698754798105922980975526542, −12.80478857988301570765718103827, −11.64086311981606842052669640661, −10.90651170949896953266886678328, −9.225887756471867024363643095451, −7.53110852120424845302532508775, −5.74558271295760522036029094996, −5.29416682557964310641477744424, −3.75287453941230953494761243342, −1.62367241305216175713329889097,
1.62367241305216175713329889097, 3.75287453941230953494761243342, 5.29416682557964310641477744424, 5.74558271295760522036029094996, 7.53110852120424845302532508775, 9.225887756471867024363643095451, 10.90651170949896953266886678328, 11.64086311981606842052669640661, 12.80478857988301570765718103827, 13.83698754798105922980975526542
