| L(s) = 1 | − 16.7i·3-s − 10.1·5-s + 80.4i·7-s − 198.·9-s − 21.6i·11-s − 179.·13-s + 170. i·15-s + 148.·17-s + 322. i·19-s + 1.34e3·21-s + 327. i·23-s − 521.·25-s + 1.96e3i·27-s + 699.·29-s − 1.04e3i·31-s + ⋯ |
| L(s) = 1 | − 1.85i·3-s − 0.407·5-s + 1.64i·7-s − 2.44·9-s − 0.178i·11-s − 1.05·13-s + 0.756i·15-s + 0.513·17-s + 0.893i·19-s + 3.04·21-s + 0.618i·23-s − 0.833·25-s + 2.69i·27-s + 0.831·29-s − 1.08i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.232706 + 0.232706i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.232706 + 0.232706i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 16.7iT - 81T^{2} \) |
| 5 | \( 1 + 10.1T + 625T^{2} \) |
| 7 | \( 1 - 80.4iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 21.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 179.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 148.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 322. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 327. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 699.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.04e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.14e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 792.T + 2.82e6T^{2} \) |
| 43 | \( 1 + 231. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 2.96e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.67e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.05e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.91e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 4.62e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 2.54e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.92e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 4.35e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 5.65e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 1.22e4T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.19e3T + 8.85e7T^{2} \) |
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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
−12.47517566075294529739423396696, −12.22161434028413676901839732408, −11.42319265170028632592699887221, −9.488224021646617720465208537008, −8.248572100802186896173556988728, −7.64011785724865443974951128040, −6.30616996198228122187872259639, −5.44864358759094108306031759714, −2.88733074828971237800980330110, −1.75574935738033641217112227385,
0.13155007511713262932038406109, 3.23000683719272320019789671514, 4.29486047088402192944632318423, 5.03805495230240749734765795242, 6.95372932195790037148888809621, 8.251246466668516608093012564254, 9.592115030251058788212217752458, 10.28052942796981764494799993432, 10.96240825826345480324814989110, 12.11333181885362946084655418628
