L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−1.94 + 3.36i)5-s + (0.343 + 2.62i)7-s − 0.999i·8-s + 3.89i·10-s + (−3.41 + 1.97i)11-s + (2.46 + 1.42i)13-s + (1.60 + 2.09i)14-s + (−0.5 − 0.866i)16-s − 0.742·17-s − 1.78i·19-s + (1.94 + 3.36i)20-s + (−1.97 + 3.41i)22-s + (5.41 + 3.12i)23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.870 + 1.50i)5-s + (0.130 + 0.991i)7-s − 0.353i·8-s + 1.23i·10-s + (−1.03 + 0.594i)11-s + (0.684 + 0.395i)13-s + (0.430 + 0.561i)14-s + (−0.125 − 0.216i)16-s − 0.179·17-s − 0.409i·19-s + (0.435 + 0.753i)20-s + (−0.420 + 0.728i)22-s + (1.12 + 0.651i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24460 + 0.839857i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24460 + 0.839857i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.343 - 2.62i)T \) |
good | 5 | \( 1 + (1.94 - 3.36i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3.41 - 1.97i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.46 - 1.42i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 0.742T + 17T^{2} \) |
| 19 | \( 1 + 1.78iT - 19T^{2} \) |
| 23 | \( 1 + (-5.41 - 3.12i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.50 + 1.44i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.04 - 1.75i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.00T + 37T^{2} \) |
| 41 | \( 1 + (-5.24 + 9.08i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.471 - 0.816i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.09 + 1.89i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (0.0105 - 0.0183i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.13 - 1.23i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.72 - 11.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 1.94iT - 71T^{2} \) |
| 73 | \( 1 - 4.85iT - 73T^{2} \) |
| 79 | \( 1 + (1.81 + 3.14i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.02 - 6.98i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 9.26T + 89T^{2} \) |
| 97 | \( 1 + (-16.2 + 9.40i)T + (48.5 - 84.0i)T^{2} \) |
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
−11.42915452530896223609279262521, −10.94056591247946066033562600502, −10.03245101324869017612689731076, −8.760247018519908859011829823387, −7.57737428390823978822554812220, −6.78173325086039552133635379516, −5.70726721909215479324829085440, −4.47649561842543947039033269205, −3.19497013275602365911151996660, −2.39672813774276824402764573634,
0.863861593254812052036981271544, 3.26757020302992888376652215260, 4.41328171508501225090197006710, 5.03945394937336288604653646817, 6.27546792186199646055931835052, 7.72694092077329712978329881853, 8.093101909292871882060196374250, 9.064636537555501506365106038059, 10.54378522218906752219305166453, 11.27948502878168632741688982839