L(s) = 1 | + (0.633 − 2.36i)3-s + (−1.73 − i)4-s + (−2 − 2i)7-s + (−2.59 − 1.50i)9-s − 11-s + (−3.46 + 3.46i)12-s + (−2.36 + 0.633i)13-s + (1.99 + 3.46i)16-s + (−1.09 + 4.09i)17-s + (4.33 + 0.5i)19-s + (−6 + 3.46i)21-s + (−1.46 − 5.46i)23-s + (1.46 + 5.46i)28-s + (−0.866 + 1.5i)29-s − 8.66i·31-s + ⋯ |
L(s) = 1 | + (0.366 − 1.36i)3-s + (−0.866 − 0.5i)4-s + (−0.755 − 0.755i)7-s + (−0.866 − 0.500i)9-s − 0.301·11-s + (−0.999 + 0.999i)12-s + (−0.656 + 0.175i)13-s + (0.499 + 0.866i)16-s + (−0.266 + 0.993i)17-s + (0.993 + 0.114i)19-s + (−1.30 + 0.755i)21-s + (−0.305 − 1.13i)23-s + (0.276 + 1.03i)28-s + (−0.160 + 0.278i)29-s − 1.55i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0891223 + 0.730423i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0891223 + 0.730423i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-4.33 - 0.5i)T \) |
good | 2 | \( 1 + (1.73 + i)T^{2} \) |
| 3 | \( 1 + (-0.633 + 2.36i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (2 + 2i)T + 7iT^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + (2.36 - 0.633i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (1.09 - 4.09i)T + (-14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (1.46 + 5.46i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.866 - 1.5i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 + (-3.46 + 3.46i)T - 37iT^{2} \) |
| 41 | \( 1 + (9 - 5.19i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (10.9 + 2.92i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (6.83 - 1.83i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.36 + 0.633i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.866 + 1.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.633 - 2.36i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-7.5 + 4.33i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.73 - 0.732i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (2.59 + 4.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + (-6.06 + 10.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.46 + 2.53i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.24140581079916300502676285266, −9.743750196454801994008859321497, −8.533796675976933018077933190842, −7.82079323609303669575528495344, −6.83494534823652798190123119286, −6.08056953610029820667500758631, −4.74486587098875870812952531646, −3.47934768702709449299094979933, −1.91657498685519080018482092223, −0.43141828203135188143038543647,
2.94367650037530551376972800908, 3.53443123454865740533719985096, 4.89335176815721131915763373119, 5.32250508128364480329283485801, 7.02030675873192717083941645936, 8.211023648088661629712485425069, 9.061798364428990082654800104939, 9.691657536988562372461563633722, 10.09273626010232836040244356090, 11.54473676706964453656355374718