L(s) = 1 | − 3.60·2-s + 0.377·3-s + 9.02·4-s + 7.18i·5-s − 1.36·6-s + (2.11 − 6.67i)7-s − 18.1·8-s − 8.85·9-s − 25.9i·10-s − 13.5i·11-s + 3.40·12-s + 11.7·13-s + (−7.63 + 24.0i)14-s + 2.71i·15-s + 29.2·16-s − 31.2·17-s + ⋯ |
L(s) = 1 | − 1.80·2-s + 0.125·3-s + 2.25·4-s + 1.43i·5-s − 0.226·6-s + (0.302 − 0.953i)7-s − 2.26·8-s − 0.984·9-s − 2.59i·10-s − 1.23i·11-s + 0.283·12-s + 0.902·13-s + (−0.545 + 1.71i)14-s + 0.180i·15-s + 1.83·16-s − 1.83·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.444972 - 0.282966i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.444972 - 0.282966i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-2.11 + 6.67i)T \) |
| 41 | \( 1 + (-30.1 + 27.8i)T \) |
good | 2 | \( 1 + 3.60T + 4T^{2} \) |
| 3 | \( 1 - 0.377T + 9T^{2} \) |
| 5 | \( 1 - 7.18iT - 25T^{2} \) |
| 11 | \( 1 + 13.5iT - 121T^{2} \) |
| 13 | \( 1 - 11.7T + 169T^{2} \) |
| 17 | \( 1 + 31.2T + 289T^{2} \) |
| 19 | \( 1 - 15.4T + 361T^{2} \) |
| 23 | \( 1 - 12.7T + 529T^{2} \) |
| 29 | \( 1 + 8.50iT - 841T^{2} \) |
| 31 | \( 1 + 39.3iT - 961T^{2} \) |
| 37 | \( 1 + 5.11T + 1.36e3T^{2} \) |
| 43 | \( 1 - 21.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 34.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 41.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 27.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 77.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 87.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 60.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 89.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 56.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 52.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 78.5T + 7.92e3T^{2} \) |
| 97 | \( 1 + 71.6T + 9.40e3T^{2} \) |
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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
−11.14869961857449512121702840409, −10.69025909462962868105047249973, −9.446140863140409866860355241772, −8.564281284118883396099706977127, −7.75171933145729513416057717951, −6.79984370992641673280446522738, −6.07146660884018158857065698117, −3.51944912612774369080284189741, −2.38658306217822092479941500966, −0.49608158604154862629116971461,
1.28105420216767406211358523293, 2.48404783872476221236755768329, 4.78518014896082379579507852657, 6.00898432343387468024152011328, 7.30502201313772336382327755561, 8.478980503087262102711060851318, 8.888341279915584376806464121296, 9.324199424889711127637845062333, 10.70217637383206476433783273921, 11.58069716545214101382981981771