L(s) = 1 | − 1.69·2-s + 0.888·4-s + (1.79 − 3.10i)5-s + 1.88·8-s + (−3.04 + 5.28i)10-s + (−1.40 − 2.43i)11-s + (0.5 + 0.866i)13-s − 4.98·16-s + (2.05 − 3.56i)17-s + (−0.444 − 0.769i)19-s + (1.59 − 2.76i)20-s + (2.38 + 4.13i)22-s + (2.93 − 5.08i)23-s + (−3.93 − 6.82i)25-s + (−0.849 − 1.47i)26-s + ⋯ |
L(s) = 1 | − 1.20·2-s + 0.444·4-s + (0.802 − 1.38i)5-s + 0.667·8-s + (−0.964 + 1.67i)10-s + (−0.423 − 0.733i)11-s + (0.138 + 0.240i)13-s − 1.24·16-s + (0.498 − 0.863i)17-s + (−0.101 − 0.176i)19-s + (0.356 − 0.617i)20-s + (0.509 + 0.882i)22-s + (0.612 − 1.06i)23-s + (−0.787 − 1.36i)25-s + (−0.166 − 0.288i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8211911215\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8211911215\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.69T + 2T^{2} \) |
| 5 | \( 1 + (-1.79 + 3.10i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.40 + 2.43i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.5 - 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.05 + 3.56i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.444 + 0.769i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.93 + 5.08i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.849 - 1.47i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.98T + 31T^{2} \) |
| 37 | \( 1 + (2.38 + 4.12i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.70 - 4.68i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.60 - 4.51i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 2.66T + 47T^{2} \) |
| 53 | \( 1 + (0.0618 - 0.107i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 8.87T + 59T^{2} \) |
| 61 | \( 1 + 3.87T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 - 2.87T + 71T^{2} \) |
| 73 | \( 1 + (5.32 - 9.21i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 7.08T + 79T^{2} \) |
| 83 | \( 1 + (-2.05 + 3.56i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.80 + 8.32i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.66 + 6.34i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.360484662635537100473894792788, −8.536550188975391711020078838653, −8.224099465232533369911917912350, −7.11133939099677762733120390355, −6.05957871843619356938303999539, −5.08144259704426812142944922539, −4.47438997810788147694569975501, −2.76513015153184071339578959636, −1.45180098131268483888199772694, −0.55770895285854709590045493652,
1.50135411808745831132655887918, 2.47579428703840871778787577310, 3.61063158211119605973804945418, 4.98877602546986830548731473105, 6.04859590140116247084695255465, 6.86677367236225550459336347235, 7.57238693415292751789874737288, 8.271786576130607640449707860534, 9.334115246666865953805535159009, 9.945604319821775548636555222289