| L(s) = 1 | − 2.61·2-s − 8.85·3-s − 1.14·4-s + 13.5·5-s + 23.1·6-s + 2.58·7-s + 23.9·8-s + 51.4·9-s − 35.3·10-s − 44.9·11-s + 10.1·12-s + 51.9·13-s − 6.77·14-s − 119.·15-s − 53.5·16-s + 75.1·17-s − 134.·18-s + 90.9·19-s − 15.4·20-s − 22.9·21-s + 117.·22-s + 64.7·23-s − 212.·24-s + 57.7·25-s − 135.·26-s − 216.·27-s − 2.96·28-s + ⋯ |
| L(s) = 1 | − 0.925·2-s − 1.70·3-s − 0.143·4-s + 1.20·5-s + 1.57·6-s + 0.139·7-s + 1.05·8-s + 1.90·9-s − 1.11·10-s − 1.23·11-s + 0.243·12-s + 1.10·13-s − 0.129·14-s − 2.06·15-s − 0.836·16-s + 1.07·17-s − 1.76·18-s + 1.09·19-s − 0.172·20-s − 0.238·21-s + 1.14·22-s + 0.586·23-s − 1.80·24-s + 0.462·25-s − 1.02·26-s − 1.54·27-s − 0.0199·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5540721142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5540721142\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 47 | \( 1 + 47T \) |
| good | 2 | \( 1 + 2.61T + 8T^{2} \) |
| 3 | \( 1 + 8.85T + 27T^{2} \) |
| 5 | \( 1 - 13.5T + 125T^{2} \) |
| 7 | \( 1 - 2.58T + 343T^{2} \) |
| 11 | \( 1 + 44.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 51.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 75.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 64.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 227.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 321.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 238.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 13.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 292.T + 7.95e4T^{2} \) |
| 53 | \( 1 - 181.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 753.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 20.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 152.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 891.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 110.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 318.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 257.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 770.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.86252096368497090115637132050, −13.75118413308948852792620647127, −12.88212180404963701903265795443, −11.27689699240746687321268083210, −10.36538622601404249194615848425, −9.561718882614230251212670891353, −7.72752997699710756367582599665, −6.00798484564888435856168837968, −5.08359997597025073250548882444, −1.07175689877785122218376823754,
1.07175689877785122218376823754, 5.08359997597025073250548882444, 6.00798484564888435856168837968, 7.72752997699710756367582599665, 9.561718882614230251212670891353, 10.36538622601404249194615848425, 11.27689699240746687321268083210, 12.88212180404963701903265795443, 13.75118413308948852792620647127, 15.86252096368497090115637132050
