L(s) = 1 | + (−0.633 + 0.366i)2-s + (0.866 + 0.5i)3-s + (−0.732 + 1.26i)4-s − 0.732·6-s + (2.5 + 0.866i)7-s − 2.53i·8-s + (0.499 + 0.866i)9-s + (1.36 − 2.36i)11-s + (−1.26 + 0.732i)12-s + 5.73i·13-s + (−1.90 + 0.366i)14-s + (−0.535 − 0.928i)16-s + (5.83 + 3.36i)17-s + (−0.633 − 0.366i)18-s + (−1.23 − 2.13i)19-s + ⋯ |
L(s) = 1 | + (−0.448 + 0.258i)2-s + (0.499 + 0.288i)3-s + (−0.366 + 0.633i)4-s − 0.298·6-s + (0.944 + 0.327i)7-s − 0.896i·8-s + (0.166 + 0.288i)9-s + (0.411 − 0.713i)11-s + (−0.366 + 0.211i)12-s + 1.58i·13-s + (−0.508 + 0.0978i)14-s + (−0.133 − 0.232i)16-s + (1.41 + 0.816i)17-s + (−0.149 − 0.0862i)18-s + (−0.282 − 0.489i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0667 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0667 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.913247 + 0.976383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.913247 + 0.976383i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.5 - 0.866i)T \) |
good | 2 | \( 1 + (0.633 - 0.366i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.36 + 2.36i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5.73iT - 13T^{2} \) |
| 17 | \( 1 + (-5.83 - 3.36i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.23 + 2.13i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.09 - 0.633i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6.19T + 29T^{2} \) |
| 31 | \( 1 + (3.23 - 5.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.23 - 3.59i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 + 7.19iT - 43T^{2} \) |
| 47 | \( 1 + (1.73 - i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.26 - 4.19i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.09 + 8.83i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.30 - 1.33i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 73 | \( 1 + (-4.03 - 2.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.69 - 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9.12iT - 83T^{2} \) |
| 89 | \( 1 + (4.56 + 7.90i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 1.07iT - 97T^{2} \) |
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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
−11.09152010701813775828414127759, −9.959108262522914231139516492390, −8.959533580005255393891590484647, −8.631478163055879184555552339181, −7.71456437367338434598713672274, −6.81795350565640618371458417213, −5.43682811367613161335282733701, −4.22608444829518348100613054512, −3.43695980300812316507625698378, −1.72609792241351778106834425704,
0.975275579198860834757675919719, 2.17877235028455554693608118324, 3.75150188242948291701146128565, 5.06017043934034719462328163870, 5.81052105334269405746371238762, 7.45910403246871476877594419935, 7.902901532447817267410596539800, 8.909332077895181696510784818091, 9.836485090541936828143365437040, 10.39912358601516584707503529513