| L(s) = 1 | + 4i·2-s − 16·4-s + 141. i·7-s − 64i·8-s − 113.·11-s − 61.7i·13-s − 566.·14-s + 256·16-s + 1.67e3i·17-s + 662.·19-s − 453. i·22-s − 86.4i·23-s + 246.·26-s − 2.26e3i·28-s − 3.23e3·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + 1.09i·7-s − 0.353i·8-s − 0.282·11-s − 0.101i·13-s − 0.772·14-s + 0.250·16-s + 1.40i·17-s + 0.420·19-s − 0.199i·22-s − 0.0340i·23-s + 0.0716·26-s − 0.546i·28-s − 0.713·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4719736207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4719736207\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 141. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 113.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 61.7iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.67e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 662.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 86.4iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.02e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.69e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.52e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 498. iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.92e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.94e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.00e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.46e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.35e4iT - 8.58e9T^{2} \) |
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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
−10.82014030903025780507327490158, −9.813334894048177029917842975090, −8.897512823279703269192380026464, −8.215063503148045061733583889189, −7.25359417144321640556116154801, −6.05101155901543030112681657840, −5.52969943641685806212438393332, −4.31239713376191532820559607211, −3.02451473746637704557482845324, −1.64122946664554805181071826434,
0.11960892290186213402406381799, 1.15178191451035399823532164686, 2.54351808588769703834920611115, 3.67555807958419296563889101152, 4.61730439561757659816290346270, 5.68879894281731023273189515588, 7.15643825132269551579814400222, 7.71438817459122120453514653131, 9.103728999122340808837869367336, 9.706373465011589432988601336414
