L(s) = 1 | + (0.707 − 0.707i)2-s + (1.71 + 0.274i)3-s − 1.00i·4-s + (−2.05 − 0.893i)5-s + (1.40 − 1.01i)6-s + (2.76 + 2.76i)7-s + (−0.707 − 0.707i)8-s + (2.84 + 0.940i)9-s + (−2.08 + 0.818i)10-s − 4.37i·11-s + (0.274 − 1.71i)12-s + (−0.0858 + 0.0858i)13-s + 3.91·14-s + (−3.26 − 2.09i)15-s − 1.00·16-s + (4.90 − 4.90i)17-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (0.987 + 0.158i)3-s − 0.500i·4-s + (−0.916 − 0.399i)5-s + (0.573 − 0.414i)6-s + (1.04 + 1.04i)7-s + (−0.250 − 0.250i)8-s + (0.949 + 0.313i)9-s + (−0.658 + 0.258i)10-s − 1.32i·11-s + (0.0793 − 0.493i)12-s + (−0.0238 + 0.0238i)13-s + 1.04·14-s + (−0.841 − 0.539i)15-s − 0.250·16-s + (1.18 − 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.33444 - 0.989862i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.33444 - 0.989862i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 + (-1.71 - 0.274i)T \) |
| 5 | \( 1 + (2.05 + 0.893i)T \) |
| 19 | \( 1 - iT \) |
good | 7 | \( 1 + (-2.76 - 2.76i)T + 7iT^{2} \) |
| 11 | \( 1 + 4.37iT - 11T^{2} \) |
| 13 | \( 1 + (0.0858 - 0.0858i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.90 + 4.90i)T - 17iT^{2} \) |
| 23 | \( 1 + (-4.04 - 4.04i)T + 23iT^{2} \) |
| 29 | \( 1 + 7.73T + 29T^{2} \) |
| 31 | \( 1 + 4.80T + 31T^{2} \) |
| 37 | \( 1 + (1.15 + 1.15i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.99iT - 41T^{2} \) |
| 43 | \( 1 + (2.22 - 2.22i)T - 43iT^{2} \) |
| 47 | \( 1 + (9.37 - 9.37i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1.61 - 1.61i)T + 53iT^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + (9.57 + 9.57i)T + 67iT^{2} \) |
| 71 | \( 1 - 10.1iT - 71T^{2} \) |
| 73 | \( 1 + (3.04 - 3.04i)T - 73iT^{2} \) |
| 79 | \( 1 - 1.24iT - 79T^{2} \) |
| 83 | \( 1 + (-9.27 - 9.27i)T + 83iT^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 + (1.05 + 1.05i)T + 97iT^{2} \) |
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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
−11.01518180891995053730347024355, −9.501806782441309745261236598295, −8.925452447727962208046139610348, −8.065363067502712446413271385260, −7.39248298097016605078270835958, −5.57286591569422142239670583547, −4.94967850217664821780602697438, −3.66198749682503761870615793973, −2.94437682050826670552354692877, −1.44211815298456770253784565888,
1.74777736659390905694549753232, 3.38458837201971509234108888781, 4.13853300079050567586985872658, 4.95003477063023334495197826216, 6.72491383694653415262912662366, 7.47313142919455508045434891400, 7.82997524433880292734781242273, 8.741636157753737972011611674669, 10.05035747775627477742158487049, 10.80168194624770534854249412948