L(s) = 1 | + i·2-s + (−0.866 + 1.5i)3-s − 4-s + (−0.5 + 0.866i)5-s + (−1.5 − 0.866i)6-s + (−2.5 + 0.866i)7-s − i·8-s + (−1.5 − 2.59i)9-s + (−0.866 − 0.5i)10-s + (1.09 − 0.633i)11-s + (0.866 − 1.5i)12-s + (−3 + 1.73i)13-s + (−0.866 − 2.5i)14-s + (−0.866 − 1.5i)15-s + 16-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.499 + 0.866i)3-s − 0.5·4-s + (−0.223 + 0.387i)5-s + (−0.612 − 0.353i)6-s + (−0.944 + 0.327i)7-s − 0.353i·8-s + (−0.5 − 0.866i)9-s + (−0.273 − 0.158i)10-s + (0.331 − 0.191i)11-s + (0.249 − 0.433i)12-s + (−0.832 + 0.480i)13-s + (−0.231 − 0.668i)14-s + (−0.223 − 0.387i)15-s + 0.250·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 11 | \( 1 + (-1.09 + 0.633i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3 - 1.73i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.09 + 2.36i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (8.19 + 4.73i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.401 + 0.232i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.19iT - 31T^{2} \) |
| 37 | \( 1 + (-2.09 - 3.63i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.59 + 6.23i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + (9.29 + 5.36i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 8.19T + 59T^{2} \) |
| 61 | \( 1 + 0.928iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 1.26iT - 71T^{2} \) |
| 73 | \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 16.5T + 79T^{2} \) |
| 83 | \( 1 + (0.401 - 0.696i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.19 + 14.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.3 - 7.73i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.04459440084626268842410801047, −9.696976163937101526538702071040, −8.819501350374797024869512041280, −7.67797162887753511526762695191, −6.55035491725307089210655902052, −6.07754067367631429430677093575, −4.91080575512478723561876561556, −3.98810345368101453656271810666, −2.87972274484168293399399826024, 0,
1.48673456284610144314278423914, 2.90615923394850078885018616289, 4.09414946486629084257041971596, 5.36051672880127327013806407837, 6.18569606829290761853384528277, 7.41877468704941256557466854515, 7.936163045557615542462880159866, 9.316400136711598590675277256039, 9.924767300285042008729476236609