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
−10.39912358601516584707503529513, −9.836485090541936828143365437040, −8.909332077895181696510784818091, −7.902901532447817267410596539800, −7.45910403246871476877594419935, −5.81052105334269405746371238762, −5.06017043934034719462328163870, −3.75150188242948291701146128565, −2.17877235028455554693608118324, −0.975275579198860834757675919719,
1.72609792241351778106834425704, 3.43695980300812316507625698378, 4.22608444829518348100613054512, 5.43682811367613161335282733701, 6.81795350565640618371458417213, 7.71456437367338434598713672274, 8.631478163055879184555552339181, 8.959533580005255393891590484647, 9.959108262522914231139516492390, 11.09152010701813775828414127759