L(s) = 1 | + (−1.5 − 0.866i)3-s + (−1 − 1.73i)5-s + (−2.5 + 0.866i)7-s + (1.5 + 2.59i)9-s + (−2 + 3.46i)11-s + (−1.5 + 2.59i)13-s + 3.46i·15-s + (−3.5 − 6.06i)17-s + (−2.5 + 4.33i)19-s + (4.5 + 0.866i)21-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s − 5.19i·27-s + (0.5 + 0.866i)29-s − 3·31-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + (−0.447 − 0.774i)5-s + (−0.944 + 0.327i)7-s + (0.5 + 0.866i)9-s + (−0.603 + 1.04i)11-s + (−0.416 + 0.720i)13-s + 0.894i·15-s + (−0.848 − 1.47i)17-s + (−0.573 + 0.993i)19-s + (0.981 + 0.188i)21-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s − 0.999i·27-s + (0.0928 + 0.160i)29-s − 0.538·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 + 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 5 | \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.5 - 2.59i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.5 + 6.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + (5.5 - 9.52i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.5 + 4.33i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 7T + 59T^{2} \) |
| 61 | \( 1 - 3T + 61T^{2} \) |
| 67 | \( 1 - 13T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 9T + 79T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.5 - 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.5 - 14.7i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.91531417605403419779691848026, −10.55689329101202657812793391340, −9.662878219913839207319557658412, −8.552443764793535707587050143900, −7.29112617787487007250299519865, −6.55768890581043077752145308703, −5.21690857840584288507882888627, −4.35617907475976882257230554471, −2.23495678738140070114362636140, 0,
3.08224443917003121759885613474, 4.10171835893774217139521115615, 5.64055223211726024280230055273, 6.46454446132880625459248694753, 7.47382598407465557345766399459, 8.850537160039160845037015473376, 10.05121210160052439283654383345, 10.83591144998887698239228116061, 11.23687913171523506384582319772