L(s) = 1 | + (−1.5 − 0.866i)3-s + (1 − 1.73i)7-s + (1.5 + 2.59i)9-s + (1.5 − 2.59i)11-s + (1 + 1.73i)13-s − 3·17-s − 19-s + (−3 + 1.73i)21-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s − 5.19i·27-s + (3 − 5.19i)29-s + (−2 − 3.46i)31-s + (−4.5 + 2.59i)33-s + 4·37-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.499i)3-s + (0.377 − 0.654i)7-s + (0.5 + 0.866i)9-s + (0.452 − 0.783i)11-s + (0.277 + 0.480i)13-s − 0.727·17-s − 0.229·19-s + (−0.654 + 0.377i)21-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s − 0.999i·27-s + (0.557 − 0.964i)29-s + (−0.359 − 0.622i)31-s + (−0.783 + 0.452i)33-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.636481 - 0.758528i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.636481 - 0.758528i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3 + 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 - 11T + 73T^{2} \) |
| 79 | \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 - 10.3i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.84765896807616910356500944427, −9.763876676698285593475005277032, −8.532532932675923376215736275720, −7.79570919160403227733370295604, −6.57285375393722334313682020533, −6.22201153478333084249243393038, −4.81852302460133989798006694550, −4.02050874707423325095045891584, −2.16933233934245689267360742257, −0.65765915851346967398994091203,
1.60533341219401324620373542745, 3.37227467801038634438479297725, 4.60426906225470960398222912636, 5.34085961246141334135075659352, 6.33817614280598642113746083621, 7.21599666919465477516588905917, 8.486093944904121466651513736447, 9.342056931783938565443015015832, 10.12583989942492981187635496446, 11.08441318358576406178598130481