| L(s) = 1 | + (−0.251 + 1.39i)2-s + (−0.989 − 1.42i)3-s + (−1.87 − 0.701i)4-s + (−0.766 + 2.25i)5-s + (2.22 − 1.01i)6-s + (−2.30 − 1.77i)7-s + (1.44 − 2.43i)8-s + (−1.04 + 2.81i)9-s + (−2.95 − 1.63i)10-s + (0.303 − 4.62i)11-s + (0.857 + 3.35i)12-s + (3.82 + 3.35i)13-s + (3.04 − 2.76i)14-s + (3.97 − 1.14i)15-s + (3.01 + 2.62i)16-s + (−0.291 + 0.291i)17-s + ⋯ |
| L(s) = 1 | + (−0.178 + 0.984i)2-s + (−0.571 − 0.820i)3-s + (−0.936 − 0.350i)4-s + (−0.342 + 1.01i)5-s + (0.909 − 0.416i)6-s + (−0.872 − 0.669i)7-s + (0.511 − 0.859i)8-s + (−0.347 + 0.937i)9-s + (−0.933 − 0.517i)10-s + (0.0914 − 1.39i)11-s + (0.247 + 0.968i)12-s + (1.06 + 0.930i)13-s + (0.813 − 0.738i)14-s + (1.02 − 0.295i)15-s + (0.754 + 0.656i)16-s + (−0.0706 + 0.0706i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.766329 + 0.298080i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.766329 + 0.298080i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.251 - 1.39i)T \) |
| 3 | \( 1 + (0.989 + 1.42i)T \) |
| good | 5 | \( 1 + (0.766 - 2.25i)T + (-3.96 - 3.04i)T^{2} \) |
| 7 | \( 1 + (2.30 + 1.77i)T + (1.81 + 6.76i)T^{2} \) |
| 11 | \( 1 + (-0.303 + 4.62i)T + (-10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (-3.82 - 3.35i)T + (1.69 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.291 - 0.291i)T - 17iT^{2} \) |
| 19 | \( 1 + (1.13 - 5.69i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (-6.64 + 5.09i)T + (5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (-2.62 + 5.31i)T + (-17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (1.09 - 1.89i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.87 + 1.56i)T + (34.1 - 14.1i)T^{2} \) |
| 41 | \( 1 + (-5.26 - 6.86i)T + (-10.6 + 39.6i)T^{2} \) |
| 43 | \( 1 + (0.0412 - 0.630i)T + (-42.6 - 5.61i)T^{2} \) |
| 47 | \( 1 + (3.30 - 12.3i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.24 + 7.84i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-6.43 - 2.18i)T + (46.8 + 35.9i)T^{2} \) |
| 61 | \( 1 + (-3.80 + 7.70i)T + (-37.1 - 48.3i)T^{2} \) |
| 67 | \( 1 + (-0.630 - 9.61i)T + (-66.4 + 8.74i)T^{2} \) |
| 71 | \( 1 + (1.92 + 4.65i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (2.14 - 5.18i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-1.77 + 6.60i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (3.58 + 10.5i)T + (-65.8 + 50.5i)T^{2} \) |
| 89 | \( 1 + (3.09 + 7.46i)T + (-62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + (12.4 - 7.16i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.91608926588045196926571723464, −10.01453788093262842905320049421, −8.735893263429073441928842426143, −7.976138735040655598390716686085, −6.99014480481629787913183061334, −6.36260016632774236468701540908, −5.97057063503023283233249489872, −4.28001513816985094360264429121, −3.17703564134155108385385304085, −0.881737734934900199402130270396,
0.842639526151002020583589701828, 2.82147278609321738211893919186, 3.92841907054158274842683063098, 4.87286510453697550058552919095, 5.56342798639285043042019613480, 7.03831049203787025104661228104, 8.553195274455712022080246694312, 9.116886903182945504978286505686, 9.701123251729906222777065803861, 10.67373038642876411295577093167
