L(s) = 1 | + (0.707 − 1.22i)2-s + (−3.62 + 2.09i)3-s + (−0.999 − 1.73i)4-s + (2.74 + 1.58i)5-s + 5.91i·6-s + (−2.24 − 6.63i)7-s − 2.82·8-s + (4.24 − 7.34i)9-s + (3.87 − 2.23i)10-s + (6.62 + 11.4i)11-s + (7.24 + 4.18i)12-s − 5.49i·13-s + (−9.70 − 1.94i)14-s − 13.2·15-s + (−2.00 + 3.46i)16-s + (−11.7 + 6.77i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−1.20 + 0.696i)3-s + (−0.249 − 0.433i)4-s + (0.548 + 0.316i)5-s + 0.985i·6-s + (−0.320 − 0.947i)7-s − 0.353·8-s + (0.471 − 0.816i)9-s + (0.387 − 0.223i)10-s + (0.601 + 1.04i)11-s + (0.603 + 0.348i)12-s − 0.422i·13-s + (−0.693 − 0.138i)14-s − 0.882·15-s + (−0.125 + 0.216i)16-s + (−0.690 + 0.398i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.685400 - 0.112094i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.685400 - 0.112094i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 1.22i)T \) |
| 7 | \( 1 + (2.24 + 6.63i)T \) |
good | 3 | \( 1 + (3.62 - 2.09i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-2.74 - 1.58i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-6.62 - 11.4i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 5.49iT - 169T^{2} \) |
| 17 | \( 1 + (11.7 - 6.77i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (0.621 + 0.358i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-1.13 + 1.96i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 20.4T + 841T^{2} \) |
| 31 | \( 1 + (-21.3 + 12.3i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (32.4 - 56.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 21.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 6.48T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-41.3 - 23.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (11.0 + 19.0i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (72.5 - 41.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-57.3 - 33.1i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (46.3 + 80.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 48.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-113. + 65.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-38.1 + 66.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 107. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (145. + 83.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 25.5iT - 9.40e3T^{2} \) |
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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
−19.69417313839934162247600962104, −17.77935336065475932600477836503, −17.03975579311497346629829026250, −15.41667580609977017558094543434, −13.75515344716880117872314244192, −12.16506698103550209906162500864, −10.68917498998943417116922782835, −9.908535361913521390313667049692, −6.41286003038764665169598384020, −4.48579382903562604163237868559,
5.53199225414551369734500404159, 6.59018977696893865850250341187, 8.967667850166112942040191886460, 11.45939919926564139447325209898, 12.57998488392336310216795537851, 13.89178078181923680281174341622, 15.83087428406578860078218621466, 16.95823834259196337226855058641, 17.93144598955751673202225066245, 19.12785436954195583261576469186