| L(s) = 1 | − 7.70i·2-s − 43.3·4-s + 12.5·5-s + 63.6i·7-s + 210. i·8-s − 96.4i·10-s + (−119. + 17.3i)11-s + 194. i·13-s + 490.·14-s + 926.·16-s − 108. i·17-s − 69.0i·19-s − 542.·20-s + (133. + 922. i)22-s − 576.·23-s + ⋯ |
| L(s) = 1 | − 1.92i·2-s − 2.70·4-s + 0.501·5-s + 1.29i·7-s + 3.28i·8-s − 0.964i·10-s + (−0.989 + 0.143i)11-s + 1.15i·13-s + 2.50·14-s + 3.61·16-s − 0.374i·17-s − 0.191i·19-s − 1.35·20-s + (0.276 + 1.90i)22-s − 1.08·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.989 - 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.753344 + 0.0543918i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.753344 + 0.0543918i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (119. - 17.3i)T \) |
| good | 2 | \( 1 + 7.70iT - 16T^{2} \) |
| 5 | \( 1 - 12.5T + 625T^{2} \) |
| 7 | \( 1 - 63.6iT - 2.40e3T^{2} \) |
| 13 | \( 1 - 194. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + 108. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 69.0iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 576.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 382. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 36.6T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.79e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.88e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 319. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.29e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 857.T + 7.89e6T^{2} \) |
| 59 | \( 1 - 2.14e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 4.96e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 5.36e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 4.95e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 3.58e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 7.14e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 156. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 7.18e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 2.41e3T + 8.85e7T^{2} \) |
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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
−12.93245497779980398981993560318, −11.95742623621965161719040101217, −11.29616595351790135627415423984, −9.969371935376785447301694062834, −9.338968735490097698095773959563, −8.250027801427593217774615012533, −5.74575377867306054305316497181, −4.53026917732720980624154173613, −2.78245898423848389128760732532, −1.86180891216162356317077279208,
0.34707161230824182915477740813, 3.93766077685691568072060573072, 5.32034483117864947421694641227, 6.27161337484746620441320969076, 7.61562929123318729444398006743, 8.116091667926550985354132550849, 9.726946514333111675966587432054, 10.47599657856253046527126571103, 12.83568091154954232906118660541, 13.52714032631208483388962335596
