| L(s) = 1 | + (0.978 + 0.207i)2-s + (1.55 − 0.768i)3-s + (0.913 + 0.406i)4-s + (−0.618 − 2.14i)5-s + (1.67 − 0.429i)6-s + (−0.0993 + 0.172i)7-s + (0.809 + 0.587i)8-s + (1.81 − 2.38i)9-s + (−0.158 − 2.23i)10-s + (−6.13 − 1.30i)11-s + (1.73 − 0.0712i)12-s + (5.94 − 1.26i)13-s + (−0.132 + 0.147i)14-s + (−2.61 − 2.85i)15-s + (0.669 + 0.743i)16-s + (4.66 + 3.38i)17-s + ⋯ |
| L(s) = 1 | + (0.691 + 0.147i)2-s + (0.896 − 0.443i)3-s + (0.456 + 0.203i)4-s + (−0.276 − 0.960i)5-s + (0.685 − 0.175i)6-s + (−0.0375 + 0.0650i)7-s + (0.286 + 0.207i)8-s + (0.605 − 0.795i)9-s + (−0.0500 − 0.705i)10-s + (−1.84 − 0.393i)11-s + (0.499 − 0.0205i)12-s + (1.64 − 0.350i)13-s + (−0.0355 + 0.0394i)14-s + (−0.674 − 0.738i)15-s + (0.167 + 0.185i)16-s + (1.13 + 0.821i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.748 + 0.663i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.42632 - 0.920837i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.42632 - 0.920837i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.55 + 0.768i)T \) |
| 5 | \( 1 + (0.618 + 2.14i)T \) |
| good | 7 | \( 1 + (0.0993 - 0.172i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (6.13 + 1.30i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-5.94 + 1.26i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (-4.66 - 3.38i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.00256 + 0.00186i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (4.01 - 4.45i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.309 - 2.94i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.379 - 3.61i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.959 + 2.95i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-0.197 + 0.0420i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (0.889 - 1.53i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.345 + 3.28i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (9.46 - 6.87i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.13 + 0.665i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (1.50 + 0.320i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.711 + 6.76i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (4.04 - 2.93i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.08 - 12.5i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.329 + 3.13i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-10.4 + 4.65i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (2.85 + 8.77i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.174 - 1.66i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−11.07407141037796325048736173929, −10.12908867163137289178548646016, −8.840606007464248500918629172896, −8.056935516944337398196974514895, −7.67429844003527979710036442517, −6.05807075022137746308857589876, −5.34051493577152760397347250515, −3.91162040836229963291979570105, −3.08914741731816861176744542627, −1.45498146308298074636380626408,
2.30250185175907009317201305388, 3.18752100033119219035715721247, 4.11653362325723647334096377442, 5.32074642179628334974364138496, 6.50744704267956035210267914290, 7.70286964056559754011163781501, 8.170581033114277927064331306247, 9.689465889329917657106882996680, 10.41469048425286368636230265326, 11.00196963392341583681993874080
