L(s) = 1 | − 1.41i·2-s − 4.76i·3-s − 2.00·4-s − 6.73·6-s − 7.05·7-s + 2.82i·8-s − 13.6·9-s + 10.4i·11-s + 9.52i·12-s − 19.0i·13-s + 9.98i·14-s + 4.00·16-s − 12.8·17-s + 19.3i·18-s + 22.7i·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.58i·3-s − 0.500·4-s − 1.12·6-s − 1.00·7-s + 0.353i·8-s − 1.52·9-s + 0.950i·11-s + 0.793i·12-s − 1.46i·13-s + 0.713i·14-s + 0.250·16-s − 0.754·17-s + 1.07i·18-s + 1.19i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5367622042\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5367622042\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (22.1 - 6.14i)T \) |
good | 3 | \( 1 + 4.76iT - 9T^{2} \) |
| 7 | \( 1 + 7.05T + 49T^{2} \) |
| 11 | \( 1 - 10.4iT - 121T^{2} \) |
| 13 | \( 1 + 19.0iT - 169T^{2} \) |
| 17 | \( 1 + 12.8T + 289T^{2} \) |
| 19 | \( 1 - 22.7iT - 361T^{2} \) |
| 29 | \( 1 - 8.61T + 841T^{2} \) |
| 31 | \( 1 - 22.2T + 961T^{2} \) |
| 37 | \( 1 - 29.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 18.7T + 1.68e3T^{2} \) |
| 43 | \( 1 - 10.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 20.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 17.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 103.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 74.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 101.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 69.4T + 5.04e3T^{2} \) |
| 73 | \( 1 - 122. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 140. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 126.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 11.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 57.0T + 9.40e3T^{2} \) |
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
−9.925683098142087933969982728898, −8.590690988684428431536723215103, −7.933175691718611089169589948103, −7.15438649082589669709860224819, −6.27422566390314108613706257164, −5.56873777792663688351665148302, −4.11603231371344376770781487004, −2.94816183940131654558349748059, −2.14806305799941650736676102385, −0.994671608480703499115631268018,
0.19030475322407122976514705157, 2.67791144855602235567178308537, 3.79349304948945152245174748880, 4.37603292200644366971779183015, 5.28759356216391775404387554499, 6.33903257036886289564760169336, 6.81074962672819913052695866787, 8.314128267489346213549326123388, 8.956473169907667384069266299950, 9.536662398538229567730666843456