| L(s) = 1 | + (1.07 + 0.915i)2-s + (−1.53 + 0.384i)3-s + (0.324 + 1.97i)4-s + (−3.70 − 1.75i)5-s + (−2.00 − 0.990i)6-s + (−0.841 − 2.77i)7-s + (−1.45 + 2.42i)8-s + (−0.437 + 0.233i)9-s + (−2.39 − 5.28i)10-s + (−2.18 + 0.324i)11-s + (−1.25 − 2.90i)12-s + (−1.41 + 3.95i)13-s + (1.63 − 3.75i)14-s + (6.36 + 1.26i)15-s + (−3.78 + 1.28i)16-s + (6.74 − 1.34i)17-s + ⋯ |
| L(s) = 1 | + (0.762 + 0.647i)2-s + (−0.886 + 0.221i)3-s + (0.162 + 0.986i)4-s + (−1.65 − 0.783i)5-s + (−0.819 − 0.404i)6-s + (−0.317 − 1.04i)7-s + (−0.515 + 0.857i)8-s + (−0.145 + 0.0779i)9-s + (−0.756 − 1.67i)10-s + (−0.660 + 0.0979i)11-s + (−0.362 − 0.838i)12-s + (−0.392 + 1.09i)13-s + (0.436 − 1.00i)14-s + (1.64 + 0.326i)15-s + (−0.947 + 0.320i)16-s + (1.63 − 0.325i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.779 + 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.779 + 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0281132 - 0.0799548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0281132 - 0.0799548i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.07 - 0.915i)T \) |
| good | 3 | \( 1 + (1.53 - 0.384i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (3.70 + 1.75i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (0.841 + 2.77i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (2.18 - 0.324i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (1.41 - 3.95i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (-6.74 + 1.34i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (0.903 + 0.819i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (8.49 - 0.836i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (2.29 + 3.09i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (1.05 + 0.437i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.350 - 7.13i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (2.54 - 3.09i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (0.967 - 3.86i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (6.17 + 4.12i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (2.30 - 3.10i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (1.26 + 3.53i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (-5.53 + 9.23i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (4.17 + 2.50i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (-4.35 + 8.14i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-0.401 + 1.32i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-5.64 - 8.44i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (0.0348 + 0.708i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (10.9 + 1.07i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (5.66 - 13.6i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−12.32925455092140978523847458887, −11.87006186186099193159496659422, −11.15651344073226077169597861883, −9.823074806438770911615964730083, −8.130464206758056210577385161330, −7.71958537351270781102614789424, −6.58215475567948388684814806352, −5.21589393598021424524916954121, −4.44650355009328448880996985153, −3.54796005787482344604429102219,
0.05523313738391081246075797923, 2.83323326980251923951505176371, 3.73085097861349355870898023314, 5.36915984650215751570697653387, 5.97710878380793943461371062369, 7.31810557713341788362524986704, 8.348877724774631713550269341114, 10.09552262162330329877326253080, 10.78966248860054858860721460672, 11.70034869904521833676648810795
