| L(s) = 1 | + (−0.415 + 0.909i)3-s + (1.04 − 1.20i)5-s + (−1.78 + 1.14i)7-s + (−0.654 − 0.755i)9-s + (0.0868 − 0.603i)11-s + (5.90 + 3.79i)13-s + (0.661 + 1.44i)15-s + (3.22 + 0.948i)17-s + (0.314 − 0.0923i)19-s + (−0.301 − 2.09i)21-s + (1.94 + 4.38i)23-s + (0.350 + 2.43i)25-s + (0.959 − 0.281i)27-s + (1.40 + 0.412i)29-s + (−0.416 − 0.912i)31-s + ⋯ |
| L(s) = 1 | + (−0.239 + 0.525i)3-s + (0.466 − 0.538i)5-s + (−0.673 + 0.432i)7-s + (−0.218 − 0.251i)9-s + (0.0261 − 0.182i)11-s + (1.63 + 1.05i)13-s + (0.170 + 0.374i)15-s + (0.783 + 0.229i)17-s + (0.0721 − 0.0211i)19-s + (−0.0657 − 0.457i)21-s + (0.406 + 0.913i)23-s + (0.0701 + 0.487i)25-s + (0.184 − 0.0542i)27-s + (0.260 + 0.0765i)29-s + (−0.0748 − 0.163i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.30481 + 0.559313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.30481 + 0.559313i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-1.94 - 4.38i)T \) |
| good | 5 | \( 1 + (-1.04 + 1.20i)T + (-0.711 - 4.94i)T^{2} \) |
| 7 | \( 1 + (1.78 - 1.14i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.0868 + 0.603i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-5.90 - 3.79i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-3.22 - 0.948i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.314 + 0.0923i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.40 - 0.412i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.416 + 0.912i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (4.42 + 5.10i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.44 + 3.97i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (2.57 - 5.63i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 7.88T + 47T^{2} \) |
| 53 | \( 1 + (-0.377 + 0.242i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-8.15 - 5.24i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-0.274 - 0.601i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.991 - 6.89i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.70 + 11.8i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (8.91 - 2.61i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (12.6 + 8.11i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-0.572 - 0.660i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-1.91 + 4.19i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-4.31 + 4.98i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.91922597385032409147486088903, −9.902531281693032754537889078740, −9.100841521018560402337060051934, −8.663705352636255432195834614690, −7.19430619792347460990678026466, −6.00652340766281590275842000489, −5.56572002218062714043095414834, −4.18732983144232855906198402736, −3.22830701758868466634059012840, −1.44051605534772571892428680935,
0.997765724026441404687777548153, 2.72304695620116421779736092273, 3.72545949995601270172666737794, 5.32616008429342626681496781275, 6.26236657344919451464001319201, 6.84161288234653408701338059587, 7.957790173355990673289735200531, 8.812863447992038895592829743168, 10.17139443384001926788361619215, 10.45365098537489212744647077814
