L(s) = 1 | + (−1.5 − 0.866i)3-s + (−3 + 1.73i)5-s + (−3 − 1.73i)7-s + (1.5 + 2.59i)9-s + (−1.5 + 2.59i)11-s + (−2 − 3.46i)13-s + 6·15-s − 1.73i·17-s + 1.73i·19-s + (3 + 5.19i)21-s + (3.5 − 6.06i)25-s − 5.19i·27-s + (−3 − 1.73i)29-s + (4.5 − 2.59i)33-s + 12·35-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + (−1.34 + 0.774i)5-s + (−1.13 − 0.654i)7-s + (0.5 + 0.866i)9-s + (−0.452 + 0.783i)11-s + (−0.554 − 0.960i)13-s + 1.54·15-s − 0.420i·17-s + 0.397i·19-s + (0.654 + 1.13i)21-s + (0.700 − 1.21i)25-s − 0.999i·27-s + (−0.557 − 0.321i)29-s + (0.783 − 0.452i)33-s + 2.02·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
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 | \( 1 + (1.5 + 0.866i)T \) |
good | 5 | \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2 + 3.46i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3 + 1.73i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.5 - 2.59i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6 - 10.3i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.5 - 4.33i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + (-3 - 1.73i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 13.8iT - 89T^{2} \) |
| 97 | \( 1 + (6.5 - 11.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.53738723894124325322831459464, −11.61037786181095395031438337570, −10.62173271796120513717245260138, −9.882101617114231288055950086789, −7.71317973727378454066988344492, −7.34671063527856289449570099151, −6.21303297723190117732479373272, −4.57699756012352974507332297910, −3.10905564533405997568619502780, 0,
3.48837582993728922263359163609, 4.67338578996700525503174029078, 5.89872616341568826783990414528, 7.14890516216849682363567770269, 8.631598278225329330423059223842, 9.455304537187462249333404721519, 10.76713271626389253569414382272, 11.81104424158415606724684820311, 12.30401005689130514723881916050