L(s) = 1 | + (0.190 − 0.587i)2-s + (−0.809 − 0.587i)3-s + (1.30 + 0.951i)4-s + (−0.5 + 0.363i)6-s − 1.61·7-s + (1.80 − 1.31i)8-s + (−0.618 − 1.90i)9-s + (−0.236 + 0.726i)11-s + (−0.5 − 1.53i)12-s + (−1.5 − 4.61i)13-s + (−0.309 + 0.951i)14-s + (0.572 + 1.76i)16-s + (0.618 − 0.449i)17-s − 1.23·18-s + (4.73 − 3.44i)19-s + ⋯ |
L(s) = 1 | + (0.135 − 0.415i)2-s + (−0.467 − 0.339i)3-s + (0.654 + 0.475i)4-s + (−0.204 + 0.148i)6-s − 0.611·7-s + (0.639 − 0.464i)8-s + (−0.206 − 0.634i)9-s + (−0.0711 + 0.219i)11-s + (−0.144 − 0.444i)12-s + (−0.416 − 1.28i)13-s + (−0.0825 + 0.254i)14-s + (0.143 + 0.440i)16-s + (0.149 − 0.108i)17-s − 0.291·18-s + (1.08 − 0.789i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0627 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.965537 - 1.02819i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.965537 - 1.02819i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 + (-0.190 + 0.587i)T + (-1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (0.809 + 0.587i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + 1.61T + 7T^{2} \) |
| 11 | \( 1 + (0.236 - 0.726i)T + (-8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (1.5 + 4.61i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.618 + 0.449i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-4.73 + 3.44i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.54 + 7.83i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 0.812i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.42 + 1.76i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.30 - 4.02i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.61 + 4.97i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 1.85T + 43T^{2} \) |
| 47 | \( 1 + (-1.30 - 0.951i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (4.42 + 3.21i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (1.28 + 3.94i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (1.45 - 4.47i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (7.47 - 5.42i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (-3.54 - 2.57i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.78 - 8.55i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (2.5 + 1.81i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-1.42 + 1.03i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-2.76 + 8.50i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (2.30 + 1.67i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.49283303654841781508477535295, −9.764670448204238385720995690309, −8.606454790413658331169354507015, −7.51161024411652820777477064799, −6.82454877370410490488302466044, −5.99579697166601094812785534756, −4.80156887983899115346476978953, −3.33388186157779623172823046162, −2.67722229788632141940645884387, −0.798266668843000338337275163586,
1.68837822592212640617476188535, 3.13456194935943319623822520212, 4.60854316481965402877492505765, 5.50272848154007032082204528360, 6.20819179822519816182768115659, 7.19907336925244671578661366406, 7.919947975287283698157251271931, 9.372946596541191482463740102350, 9.949325877433115236286541900790, 10.91524553008624257674536926254