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 | \( 1 + (-16.2 - 9.40i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.27948502878168632741688982839, −10.54378522218906752219305166453, −9.064636537555501506365106038059, −8.093101909292871882060196374250, −7.72694092077329712978329881853, −6.27546792186199646055931835052, −5.03945394937336288604653646817, −4.41328171508501225090197006710, −3.26757020302992888376652215260, −0.863861593254812052036981271544,
2.39672813774276824402764573634, 3.19497013275602365911151996660, 4.47649561842543947039033269205, 5.70726721909215479324829085440, 6.78173325086039552133635379516, 7.57737428390823978822554812220, 8.760247018519908859011829823387, 10.03245101324869017612689731076, 10.94056591247946066033562600502, 11.42915452530896223609279262521