| L(s) = 1 | + (0.622 + 8.97i)3-s − 0.562i·5-s − 55.8·7-s + (−80.2 + 11.1i)9-s − 87.7i·11-s − 71.5·13-s + (5.04 − 0.349i)15-s − 334. i·17-s + 365.·19-s + (−34.7 − 501. i)21-s + 789. i·23-s + 624.·25-s + (−150. − 713. i)27-s − 236. i·29-s + 1.28e3·31-s + ⋯ |
| L(s) = 1 | + (0.0691 + 0.997i)3-s − 0.0224i·5-s − 1.13·7-s + (−0.990 + 0.138i)9-s − 0.725i·11-s − 0.423·13-s + (0.0224 − 0.00155i)15-s − 1.15i·17-s + 1.01·19-s + (−0.0788 − 1.13i)21-s + 1.49i·23-s + 0.999·25-s + (−0.206 − 0.978i)27-s − 0.281i·29-s + 1.33·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0691i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.997 - 0.0691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.424429004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.424429004\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.622 - 8.97i)T \) |
| good | 5 | \( 1 + 0.562iT - 625T^{2} \) |
| 7 | \( 1 + 55.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + 87.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 71.5T + 2.85e4T^{2} \) |
| 17 | \( 1 + 334. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 365.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 789. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 236. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.28e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 513.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.18e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 204.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.31e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 317. iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.27e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.77e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 7.24e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 4.83e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 6.98e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 9.33e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 838. iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.39e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 6.57e3T + 8.85e7T^{2} \) |
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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
−10.62457035887665844005960766851, −9.524215683285161445877539175154, −9.376924373967168231620698955483, −8.051338224933779466615220035589, −6.87470616766135690271822770769, −5.73841812207726756092745976868, −4.86519313085676306503993409988, −3.47586118182730686994019014032, −2.85001785160331136548113491872, −0.55187675016057397219342011924,
0.847477951099861828787223928241, 2.32099140579858166388112946448, 3.35213722912220164683890750417, 4.90514051009526220186127753408, 6.28848397440935273444100435797, 6.76792362964661772249042830880, 7.82177867233517157789345942488, 8.774832145098360077878356120075, 9.792618880601765567958999808245, 10.63867424672818914466526964798
