| L(s) = 1 | + 2i·2-s − 4·4-s + 5.92·5-s + (0.757 + 18.5i)7-s − 8i·8-s + 11.8i·10-s − 0.301i·11-s + 72.5i·13-s + (−37.0 + 1.51i)14-s + 16·16-s − 44.3·17-s + 84.4i·19-s − 23.7·20-s + 0.603·22-s + 63.2i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 0.530·5-s + (0.0408 + 0.999i)7-s − 0.353i·8-s + 0.374i·10-s − 0.00826i·11-s + 1.54i·13-s + (−0.706 + 0.0289i)14-s + 0.250·16-s − 0.633·17-s + 1.01i·19-s − 0.265·20-s + 0.00584·22-s + 0.573i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.610 - 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.636820 + 1.29441i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.636820 + 1.29441i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-0.757 - 18.5i)T \) |
| good | 5 | \( 1 - 5.92T + 125T^{2} \) |
| 11 | \( 1 + 0.301iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 72.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 44.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 84.4iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 63.2iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 183. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 50.3iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 285.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 216.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 14.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 423.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 202. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 734.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 633. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 133.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.04e3iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 152. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 819.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 583.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 409. iT - 9.12e5T^{2} \) |
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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
−13.51054783153391551323757341482, −12.29899956586532037804979899439, −11.34491982500148883340445560336, −9.725869175423617765414807426205, −9.079447780674658967911090666642, −7.892496441322687572476169371544, −6.47670093130292118147863630919, −5.66349153820914266934505209581, −4.20767686746006861525271142865, −2.08618422004324644028870084577,
0.76800914719066119922964368984, 2.66428078967646497601832335950, 4.20019020871492822204788178453, 5.57077225089392050035548386401, 7.11137367269576773062238258262, 8.384636230473495473665358653072, 9.660630771980109104309440295242, 10.52805273841549467320084752876, 11.27050310816421818229840870376, 12.78214427054473735162871506207
