L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (−2.23 − 0.0685i)5-s + 1.00·6-s + (−2.16 + 2.16i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (1.53 + 1.62i)10-s − 5.68·11-s + (−0.707 − 0.707i)12-s + (3.92 − 3.92i)13-s + 3.06·14-s + (1.62 − 1.53i)15-s − 1.00·16-s + (4.99 − 4.99i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (−0.999 − 0.0306i)5-s + 0.408·6-s + (−0.818 + 0.818i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.484 + 0.515i)10-s − 1.71·11-s + (−0.204 − 0.204i)12-s + (1.08 − 1.08i)13-s + 0.818·14-s + (0.420 − 0.395i)15-s − 0.250·16-s + (1.21 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.510852 - 0.273024i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.510852 - 0.273024i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.23 + 0.0685i)T \) |
| 19 | \( 1 + (-4.01 - 1.68i)T \) |
good | 7 | \( 1 + (2.16 - 2.16i)T - 7iT^{2} \) |
| 11 | \( 1 + 5.68T + 11T^{2} \) |
| 13 | \( 1 + (-3.92 + 3.92i)T - 13iT^{2} \) |
| 17 | \( 1 + (-4.99 + 4.99i)T - 17iT^{2} \) |
| 23 | \( 1 + (-4.30 - 4.30i)T + 23iT^{2} \) |
| 29 | \( 1 + 4.04T + 29T^{2} \) |
| 31 | \( 1 + 4.04iT - 31T^{2} \) |
| 37 | \( 1 + (-6.19 - 6.19i)T + 37iT^{2} \) |
| 41 | \( 1 + 6.38iT - 41T^{2} \) |
| 43 | \( 1 + (4.15 + 4.15i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.24 + 3.24i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.40 + 5.40i)T - 53iT^{2} \) |
| 59 | \( 1 + 2.39T + 59T^{2} \) |
| 61 | \( 1 + 2.33T + 61T^{2} \) |
| 67 | \( 1 + (6.12 + 6.12i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.51iT - 71T^{2} \) |
| 73 | \( 1 + (2.07 + 2.07i)T + 73iT^{2} \) |
| 79 | \( 1 - 4.23T + 79T^{2} \) |
| 83 | \( 1 + (5.30 + 5.30i)T + 83iT^{2} \) |
| 89 | \( 1 + 2.50T + 89T^{2} \) |
| 97 | \( 1 + (-6.22 - 6.22i)T + 97iT^{2} \) |
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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
−10.57409315650438746765196943835, −9.859420743837082170662982997386, −8.981009289912931830661789688742, −7.927947820794916889985967008935, −7.39600330847396918096274395757, −5.74922058890777766288664513037, −5.13656520028663550019740236382, −3.40561843656419584694602019679, −3.00133153340231192513429285392, −0.55008891708457845029654201241,
0.988382596465890019942722416085, 3.11212484915956873810260616926, 4.32004369843668816492924141573, 5.56505539216051752813687334301, 6.56498341097128497502827226911, 7.39856126732999368088403185973, 7.959440831343749185252985459850, 8.941451625482311712102243704908, 10.17389168254825680230747309580, 10.77460155187317416846703119860