| L(s) = 1 | + (−0.173 − 0.984i)2-s + (−1.61 − 0.636i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)5-s + (−0.347 + 1.69i)6-s + (−1.33 − 0.487i)7-s + (0.5 + 0.866i)8-s + (2.18 + 2.05i)9-s + (−0.5 + 0.866i)10-s + (−4.46 + 3.74i)11-s + (1.73 + 0.0471i)12-s + (0.536 − 3.04i)13-s + (−0.247 + 1.40i)14-s + (0.824 + 1.52i)15-s + (0.766 − 0.642i)16-s + (−2.83 + 4.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.930 − 0.367i)3-s + (−0.469 + 0.171i)4-s + (−0.342 − 0.287i)5-s + (−0.141 + 0.692i)6-s + (−0.506 − 0.184i)7-s + (0.176 + 0.306i)8-s + (0.729 + 0.683i)9-s + (−0.158 + 0.273i)10-s + (−1.34 + 1.12i)11-s + (0.499 + 0.0135i)12-s + (0.148 − 0.844i)13-s + (−0.0661 + 0.375i)14-s + (0.212 + 0.393i)15-s + (0.191 − 0.160i)16-s + (−0.688 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0751284 + 0.0874351i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0751284 + 0.0874351i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 3 | \( 1 + (1.61 + 0.636i)T \) |
| 5 | \( 1 + (0.766 + 0.642i)T \) |
| good | 7 | \( 1 + (1.33 + 0.487i)T + (5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (4.46 - 3.74i)T + (1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.536 + 3.04i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (2.83 - 4.91i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.33 - 5.78i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.85 - 0.676i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.64 + 9.33i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (9.07 - 3.30i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.632 + 1.09i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.02 - 5.79i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-5.51 + 4.62i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (11.4 + 4.15i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 4.43T + 53T^{2} \) |
| 59 | \( 1 + (-0.776 - 0.651i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.944 - 0.343i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.281 - 1.59i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-0.883 + 1.52i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.07 + 3.60i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.116 + 0.659i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (0.504 + 2.85i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-3.75 - 6.50i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.89 - 3.27i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−12.29480290748265587666702609701, −11.23200739596253467014976377917, −10.32422293020502652280385550242, −9.826811286635275296946900521360, −8.119479406731869810927838084291, −7.49812828191326831979583185986, −6.01083474400400251273311607299, −5.01449992012632809453623893440, −3.75508239914916094308882086533, −1.91270468699858006144612687575,
0.099501661338913624632052161730, 3.19528257883400533845699076989, 4.74160181725105677904053704861, 5.59375070088240625955662183381, 6.69057472949395560600785832943, 7.45400606360089894551340654951, 8.908817604369631182214380013349, 9.626852305817289347321487548774, 11.02489220388357464626355031086, 11.24731741367562079563250152239
