| L(s) = 1 | + (0.978 − 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (1.42 − 1.58i)5-s + (−0.309 − 0.951i)6-s + (0.588 − 2.57i)7-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (1.06 − 1.85i)10-s + (−3.15 + 1.02i)11-s + (−0.5 − 0.866i)12-s + (−0.776 + 2.39i)13-s + (0.0391 − 2.64i)14-s + (−1.72 − 1.25i)15-s + (0.669 − 0.743i)16-s + (6.84 + 1.45i)17-s + ⋯ |
| L(s) = 1 | + (0.691 − 0.147i)2-s + (−0.0603 − 0.574i)3-s + (0.456 − 0.203i)4-s + (0.639 − 0.710i)5-s + (−0.126 − 0.388i)6-s + (0.222 − 0.974i)7-s + (0.286 − 0.207i)8-s + (−0.326 + 0.0693i)9-s + (0.337 − 0.585i)10-s + (−0.951 + 0.308i)11-s + (−0.144 − 0.250i)12-s + (−0.215 + 0.663i)13-s + (0.0104 − 0.707i)14-s + (−0.446 − 0.324i)15-s + (0.167 − 0.185i)16-s + (1.65 + 0.352i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0721 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0721 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.60883 - 1.49664i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60883 - 1.49664i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.978 + 0.207i)T \) |
| 3 | \( 1 + (0.104 + 0.994i)T \) |
| 7 | \( 1 + (-0.588 + 2.57i)T \) |
| 11 | \( 1 + (3.15 - 1.02i)T \) |
| good | 5 | \( 1 + (-1.42 + 1.58i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (0.776 - 2.39i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-6.84 - 1.45i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (5.60 + 2.49i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (1.08 + 1.88i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.459 + 0.333i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-6.82 - 7.57i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.622 + 5.92i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (-5.91 + 4.29i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 2.19T + 43T^{2} \) |
| 47 | \( 1 + (-5.58 - 2.48i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-3.32 - 3.69i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-2.34 + 1.04i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (10.2 - 11.4i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-0.164 + 0.284i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.47 + 4.54i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (0.117 - 0.0522i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (15.7 - 3.34i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-2.06 - 6.34i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.49 - 11.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.0430 - 0.132i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.74029753910886430129437701477, −10.26778032754825656319057378970, −9.035561064458177653795860945292, −7.891512150521486228366360257991, −7.10161251079092069907356304228, −6.00381253555755589742504866251, −5.07413103930086287295918758085, −4.13713048535411038899590015679, −2.52470034735702844044071824256, −1.22874305601994261902183046076,
2.40360140435429677202088441320, 3.17441853073450438439486139095, 4.67471966151215073906517243879, 5.77652164119124011522825265633, 6.04769636245255155466520406473, 7.66614621641818813613051360946, 8.392651559892021260125489469504, 9.842933309962536120785173282547, 10.28893336098065928721627580479, 11.28238831517754574348090040152
