| L(s) = 1 | + (−0.221 − 1.39i)2-s + (−1.54 − 0.786i)3-s + (−1.90 + 0.618i)4-s + (−4.84 + 1.25i)5-s + (−0.756 + 2.32i)6-s + (3.10 + 3.10i)7-s + (1.28 + 2.52i)8-s + (1.76 + 2.42i)9-s + (2.82 + 6.48i)10-s + (4.85 + 3.52i)11-s + (3.42 + 0.541i)12-s + (−3.04 + 19.2i)13-s + (3.64 − 5.02i)14-s + (8.45 + 1.86i)15-s + (3.23 − 2.35i)16-s + (−4.84 + 2.46i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 − 0.698i)2-s + (−0.514 − 0.262i)3-s + (−0.475 + 0.154i)4-s + (−0.968 + 0.250i)5-s + (−0.126 + 0.388i)6-s + (0.443 + 0.443i)7-s + (0.160 + 0.315i)8-s + (0.195 + 0.269i)9-s + (0.282 + 0.648i)10-s + (0.440 + 0.320i)11-s + (0.285 + 0.0451i)12-s + (−0.234 + 1.47i)13-s + (0.260 − 0.358i)14-s + (0.563 + 0.124i)15-s + (0.202 − 0.146i)16-s + (−0.284 + 0.145i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.531 - 0.846i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.531 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.581921 + 0.321637i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.581921 + 0.321637i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.221 + 1.39i)T \) |
| 3 | \( 1 + (1.54 + 0.786i)T \) |
| 5 | \( 1 + (4.84 - 1.25i)T \) |
| good | 7 | \( 1 + (-3.10 - 3.10i)T + 49iT^{2} \) |
| 11 | \( 1 + (-4.85 - 3.52i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (3.04 - 19.2i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (4.84 - 2.46i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (-15.7 - 5.12i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (39.8 - 6.31i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (4.59 - 1.49i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (15.6 - 48.2i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-2.55 - 0.404i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-12.5 + 9.09i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (46.2 - 46.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-15.6 + 30.6i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (66.2 + 33.7i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-28.7 - 39.5i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-3.60 - 2.62i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (40.2 - 20.5i)T + (2.63e3 - 3.63e3i)T^{2} \) |
| 71 | \( 1 + (2.75 + 8.46i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-85.7 + 13.5i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-64.3 + 20.9i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-69.2 - 135. i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-31.2 + 43.0i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-85.7 + 168. i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−12.43548562454326141702729503102, −11.80382953868295774105650072259, −11.32579050109016453102704887323, −10.05579587441195043657041138284, −8.868673409589343072533047172911, −7.73446768588034329554578716217, −6.59436343384455138821257873353, −4.92183707584770218251432264675, −3.77096834423983450630842602103, −1.81813296623370171580470117395,
0.48800965520734897883737655292, 3.73053862680985914602276981559, 4.87186385493568643004827475490, 6.04025065692338407202841938458, 7.50828897492575980490134820525, 8.108040540851968418811209930350, 9.474672626897042439724949406713, 10.64169630931096806027147785604, 11.58370495639631795945107630642, 12.55621486909962804110287137393
