| L(s) = 1 | + (−2.59 + 1.5i)5-s + (2.5 − 4.33i)7-s + (−12.9 − 7.5i)11-s + (5 + 8.66i)13-s + 18i·17-s + 16·19-s + (10.3 − 6i)23-s + (−8 + 13.8i)25-s + (−25.9 − 15i)29-s + (−0.5 − 0.866i)31-s + 15.0i·35-s + 20·37-s + (51.9 − 30i)41-s + (25 − 43.3i)43-s + (−5.19 − 3i)47-s + ⋯ |
| L(s) = 1 | + (−0.519 + 0.300i)5-s + (0.357 − 0.618i)7-s + (−1.18 − 0.681i)11-s + (0.384 + 0.666i)13-s + 1.05i·17-s + 0.842·19-s + (0.451 − 0.260i)23-s + (−0.320 + 0.554i)25-s + (−0.895 − 0.517i)29-s + (−0.0161 − 0.0279i)31-s + 0.428i·35-s + 0.540·37-s + (1.26 − 0.731i)41-s + (0.581 − 1.00i)43-s + (−0.110 − 0.0638i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.617089060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.617089060\) |
| \(L(2)\) |
|
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 | \( 1 + (2.59 - 1.5i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-2.5 + 4.33i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (12.9 + 7.5i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-5 - 8.66i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 18iT - 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 + (-10.3 + 6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (25.9 + 15i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 20T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-51.9 + 30i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-25 + 43.3i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (5.19 + 3i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 27iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-25.9 + 15i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-38 + 65.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 90iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 65T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-7 + 12.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-2.59 - 1.5i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 90iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-42.5 + 73.6i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.439842092941892300183331430833, −8.494813256331509113083613443186, −7.74629008778868928204196858337, −7.21657316348049184493825164113, −6.06812553130097238585836383237, −5.27379475028953192727072940739, −4.10930485550061923183132569582, −3.44581586846874099441824584774, −2.15449223185052976206785960458, −0.70542325164991470253062493096,
0.78164468594421581121334830775, 2.32180989732117116202332991987, 3.21125709912444283166010026863, 4.53202311242103292732137089744, 5.21066757046961970689049725133, 5.97320617641067174411444351263, 7.44539739157736942254804321877, 7.67534565714286514358284101653, 8.651868055863429349869014549248, 9.453069804774984751718044323036
