| L(s) = 1 | − 5.47·2-s + 8.31·3-s + 21.9·4-s + 5.94·5-s − 45.4·6-s − 76.2·8-s + 42.0·9-s − 32.5·10-s − 31.3·11-s + 182.·12-s + 17.0·13-s + 49.3·15-s + 241.·16-s − 55.8·17-s − 230.·18-s + 16.2·19-s + 130.·20-s + 171.·22-s + 83.9·23-s − 633.·24-s − 89.7·25-s − 93.0·26-s + 125.·27-s − 100.·29-s − 270.·30-s + 153.·31-s − 712.·32-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 1.59·3-s + 2.74·4-s + 0.531·5-s − 3.09·6-s − 3.37·8-s + 1.55·9-s − 1.02·10-s − 0.859·11-s + 4.38·12-s + 0.362·13-s + 0.849·15-s + 3.77·16-s − 0.797·17-s − 3.01·18-s + 0.196·19-s + 1.45·20-s + 1.66·22-s + 0.761·23-s − 5.39·24-s − 0.717·25-s − 0.701·26-s + 0.893·27-s − 0.642·29-s − 1.64·30-s + 0.890·31-s − 3.93·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 5.47T + 8T^{2} \) |
| 3 | \( 1 - 8.31T + 27T^{2} \) |
| 5 | \( 1 - 5.94T + 125T^{2} \) |
| 11 | \( 1 + 31.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 17.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 55.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 16.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 83.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 153.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 98.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 330.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 340.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 79.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 56.1T + 1.48e5T^{2} \) |
| 59 | \( 1 - 196.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 555.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 379.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.04e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.20e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 937.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 598.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 988.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 890.T + 9.12e5T^{2} \) |
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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
−8.227591816047036871967280225388, −7.991658711500715625378177608068, −7.04792511248544141712807091819, −6.46042788634427490869746206298, −5.24951572849487304663816461524, −3.61268446637640131397899130944, −2.76941414879734860592404965262, −2.12679292438699690617869008125, −1.38775442567163641799642130396, 0,
1.38775442567163641799642130396, 2.12679292438699690617869008125, 2.76941414879734860592404965262, 3.61268446637640131397899130944, 5.24951572849487304663816461524, 6.46042788634427490869746206298, 7.04792511248544141712807091819, 7.991658711500715625378177608068, 8.227591816047036871967280225388
