| L(s) = 1 | − 5.19i·3-s − 2.60·5-s − 77.5i·7-s − 27·9-s + 219. i·11-s − 83.5·13-s + 13.5i·15-s − 295.·17-s − 598. i·19-s − 403.·21-s + 826. i·23-s − 618.·25-s + 140. i·27-s + 638.·29-s + 243. i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.104·5-s − 1.58i·7-s − 0.333·9-s + 1.81i·11-s − 0.494·13-s + 0.0601i·15-s − 1.02·17-s − 1.65i·19-s − 0.913·21-s + 1.56i·23-s − 0.989·25-s + 0.192i·27-s + 0.759·29-s + 0.253i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5914433876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5914433876\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19iT \) |
| good | 5 | \( 1 + 2.60T + 625T^{2} \) |
| 7 | \( 1 + 77.5iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 219. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 83.5T + 2.85e4T^{2} \) |
| 17 | \( 1 + 295.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 598. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 826. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 638.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 243. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.15e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 29.0T + 2.82e6T^{2} \) |
| 43 | \( 1 - 427. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.92e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.06e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 3.12e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 249.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 4.45e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 3.46e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.91e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 1.10e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 3.78e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.31e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.38e4T + 8.85e7T^{2} \) |
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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
−11.02831109987135207291652995808, −9.966772543365350292250184760636, −9.291063780589192662182211900405, −7.72096437352704716520476514317, −7.28700890635893561586626441552, −6.54465387743592152612900451430, −4.84016334603044814894616170184, −4.14876581160508899127589775064, −2.50986086334364139978490730965, −1.22129315775842342308822975345,
0.17265949234585761640039368343, 2.23671208170614427985279506486, 3.26681010616525890690710945022, 4.56304433677471438866026514527, 5.80994390690282925835161621704, 6.22120330451437679789782605726, 8.130219455882727572465941259247, 8.580046539343146161683956464490, 9.482142552513212543182476486807, 10.51434527783057566339726770388
