L(s) = 1 | + (−1.21 + 2.09i)5-s + (1.85 − 1.88i)7-s + (2.35 + 4.07i)11-s − 3.42·13-s + (0.851 + 1.47i)17-s + (−0.641 + 1.11i)19-s + (−0.562 + 0.974i)23-s + (−0.430 − 0.746i)25-s + 4.70·29-s + (1.71 + 2.96i)31-s + (1.72 + 6.17i)35-s + (−4.27 + 7.40i)37-s − 3.71·41-s + 5.54·43-s + (−5.91 + 10.2i)47-s + ⋯ |
L(s) = 1 | + (−0.541 + 0.937i)5-s + (0.699 − 0.714i)7-s + (0.709 + 1.22i)11-s − 0.948·13-s + (0.206 + 0.357i)17-s + (−0.147 + 0.254i)19-s + (−0.117 + 0.203i)23-s + (−0.0861 − 0.149i)25-s + 0.873·29-s + (0.307 + 0.532i)31-s + (0.290 + 1.04i)35-s + (−0.702 + 1.21i)37-s − 0.580·41-s + 0.845·43-s + (−0.862 + 1.49i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.03184 + 0.853760i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.03184 + 0.853760i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.85 + 1.88i)T \) |
good | 5 | \( 1 + (1.21 - 2.09i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.35 - 4.07i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 17 | \( 1 + (-0.851 - 1.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.641 - 1.11i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.562 - 0.974i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4.70T + 29T^{2} \) |
| 31 | \( 1 + (-1.71 - 2.96i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.27 - 7.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.71T + 41T^{2} \) |
| 43 | \( 1 - 5.54T + 43T^{2} \) |
| 47 | \( 1 + (5.91 - 10.2i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.13 - 8.88i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.06 - 3.57i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.62 + 8.01i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.56 - 9.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + (1.06 + 1.85i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.26 + 12.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8.42T + 83T^{2} \) |
| 89 | \( 1 + (-8.04 + 13.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 16.2T + 97T^{2} \) |
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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
−10.33320278997075088377117813570, −10.04354052538162876232332127744, −8.763357715636282587376731565316, −7.65730443052436874405863546215, −7.21916780906514629898219558293, −6.39899195473838974325474853011, −4.87822381829210239425774100799, −4.18770447212822073340441491606, −3.01680077458057651211961222483, −1.60217678453446610127157597266,
0.72438325538532723074550833489, 2.32976847845232707851885135218, 3.73881449858023001642753825111, 4.81744284463368629970372125983, 5.48306852314886729840614944427, 6.63708370166040662870312000845, 7.80032455774992161500963293398, 8.576920081656051125607729342771, 8.991833618436407203585080389843, 10.10320734175600489400191636719