L(s) = 1 | − 1.41i·2-s + (2.16 + 2.08i)3-s − 2.00·4-s + 2.23i·5-s + (2.94 − 3.05i)6-s + 12.7·7-s + 2.82i·8-s + (0.333 + 8.99i)9-s + 3.16·10-s + 10.4i·11-s + (−4.32 − 4.16i)12-s − 10.0·13-s − 18.0i·14-s + (−4.65 + 4.83i)15-s + 4.00·16-s − 0.278i·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.720 + 0.693i)3-s − 0.500·4-s + 0.447i·5-s + (0.490 − 0.509i)6-s + 1.82·7-s + 0.353i·8-s + (0.0370 + 0.999i)9-s + 0.316·10-s + 0.947i·11-s + (−0.360 − 0.346i)12-s − 0.776·13-s − 1.28i·14-s + (−0.310 + 0.322i)15-s + 0.250·16-s − 0.0163i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.693 - 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.556914959\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.556914959\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 + (-2.16 - 2.08i)T \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 - 4.79iT \) |
good | 7 | \( 1 - 12.7T + 49T^{2} \) |
| 11 | \( 1 - 10.4iT - 121T^{2} \) |
| 13 | \( 1 + 10.0T + 169T^{2} \) |
| 17 | \( 1 + 0.278iT - 289T^{2} \) |
| 19 | \( 1 - 14.4T + 361T^{2} \) |
| 29 | \( 1 + 1.71iT - 841T^{2} \) |
| 31 | \( 1 + 36.9T + 961T^{2} \) |
| 37 | \( 1 - 17.1T + 1.36e3T^{2} \) |
| 41 | \( 1 + 36.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 41.5T + 1.84e3T^{2} \) |
| 47 | \( 1 - 73.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 20.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 74.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 77.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 103.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 56.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 39.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 81.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 37.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 129. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 39.8T + 9.40e3T^{2} \) |
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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
−10.36541540293747522662393837177, −9.651776500437221228786768910132, −8.818057265094320907344845198183, −7.81515490197786973405177111410, −7.34264610433277445269449862834, −5.32856067553715330055977779190, −4.74345751077105553704199779954, −3.81977043160908738988768554368, −2.50450036600534289075207519403, −1.69586085765739578037691420900,
0.888526381628877877172653744697, 2.09026941195563367980541831984, 3.63284136568172973064990071381, 4.87349793502907998922824494468, 5.57684740596937164890255412846, 6.86603494795234259452130151424, 7.73757649539329693264947288323, 8.254434108399255248680018310008, 8.876778348115839527868689734785, 9.862161082123358783823083105502