| L(s) = 1 | − 2.82i·2-s − 8.00·4-s + 11.1i·5-s + 72.5·7-s + 22.6i·8-s + 31.6·10-s + 78.5i·11-s − 120.·13-s − 205. i·14-s + 64.0·16-s − 68.1i·17-s − 506.·19-s − 89.4i·20-s + 222.·22-s − 635. i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.447i·5-s + 1.48·7-s + 0.353i·8-s + 0.316·10-s + 0.648i·11-s − 0.714·13-s − 1.04i·14-s + 0.250·16-s − 0.235i·17-s − 1.40·19-s − 0.223i·20-s + 0.458·22-s − 1.20i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.964594223\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.964594223\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 11.1iT \) |
| good | 7 | \( 1 - 72.5T + 2.40e3T^{2} \) |
| 11 | \( 1 - 78.5iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 120.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 68.1iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 506.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 635. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.29e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 489.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 2.07e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 361. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.20e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 3.79e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.08e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 3.70e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.74e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 8.37e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.52e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.86e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 5.46e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 6.93e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 6.62e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 679.T + 8.85e7T^{2} \) |
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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
−9.669431907963165926727340831574, −8.585235239501802204218614963051, −7.913854576472523117366309676333, −7.00499975824717116399251689762, −5.79419952944766594478022103088, −4.61431890517131911346969010470, −4.20605322833383319771203240632, −2.49805959112862479467209864355, −2.00105195491671273906613735034, −0.52120991115950828921399846450,
0.967485611988553559092996239397, 2.11389025409273716616533529391, 3.76372035664846111599491120668, 4.77539013307807382724165733407, 5.35561046575881419579111670347, 6.37778295444346712142785654001, 7.51962528688208110243190213701, 8.112140919001255687763439562324, 8.811449892884899936430070167047, 9.648418070350310649987507662595
