L(s) = 1 | + (−3.47 + 2.00i)5-s + (2.66 + 1.54i)7-s + (2.97 − 5.15i)11-s + (0.673 + 1.16i)13-s + 4.87i·17-s − 7.49i·19-s + (−1.28 − 2.23i)23-s + (5.57 − 9.65i)25-s + (−5.60 − 3.23i)29-s + (−1.28 + 0.743i)31-s − 12.3·35-s + 1.91·37-s + (1.69 − 0.978i)41-s + (6.79 + 3.92i)43-s + (−1.14 + 1.97i)47-s + ⋯ |
L(s) = 1 | + (−1.55 + 0.898i)5-s + (1.00 + 0.582i)7-s + (0.897 − 1.55i)11-s + (0.186 + 0.323i)13-s + 1.18i·17-s − 1.71i·19-s + (−0.268 − 0.465i)23-s + (1.11 − 1.93i)25-s + (−1.04 − 0.600i)29-s + (−0.231 + 0.133i)31-s − 2.09·35-s + 0.315·37-s + (0.264 − 0.152i)41-s + (1.03 + 0.598i)43-s + (−0.166 + 0.288i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.456180010\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.456180010\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (3.47 - 2.00i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.66 - 1.54i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.97 + 5.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.673 - 1.16i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 4.87iT - 17T^{2} \) |
| 19 | \( 1 + 7.49iT - 19T^{2} \) |
| 23 | \( 1 + (1.28 + 2.23i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.60 + 3.23i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.28 - 0.743i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.91T + 37T^{2} \) |
| 41 | \( 1 + (-1.69 + 0.978i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-6.79 - 3.92i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.14 - 1.97i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 0.871iT - 53T^{2} \) |
| 59 | \( 1 + (0.650 + 1.12i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.29 + 5.71i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.44 + 4.87i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 4.64T + 71T^{2} \) |
| 73 | \( 1 + 2.14T + 73T^{2} \) |
| 79 | \( 1 + (-8.08 - 4.66i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.09 + 8.82i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 0.947iT - 89T^{2} \) |
| 97 | \( 1 + (5.15 - 8.93i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.582228563763246269494959765675, −8.213736300822043265293969485360, −7.44350454240710092321771452197, −6.56114432258285778990337578899, −5.94591583936131006565391114650, −4.73367907646800298546080191187, −3.94435541075770939724006145560, −3.31288247919827119129522140737, −2.21099339377796341361707293536, −0.65352889722238426094799615505,
0.941805073335913304249725056362, 1.85319991844255103532142991748, 3.62875539221046008838935616988, 4.12162011028401188230593149416, 4.76999209580942301902600835983, 5.54624443819288877819161281821, 6.98325634156905390522808377512, 7.61056052094179676692678426938, 7.87265394565892719651836565737, 8.837534074411726457882018666716