L(s) = 1 | + (−1 − i)2-s + (1.5 + 0.866i)3-s + 2i·4-s + (−2.23 − 0.133i)5-s + (−0.633 − 2.36i)6-s + (1.13 − 4.23i)7-s + (2 − 2i)8-s + (1.5 + 2.59i)9-s + (2.09 + 2.36i)10-s + (−1 − 1.73i)11-s + (−1.73 + 3i)12-s + (1.73 − 0.464i)13-s + (−5.36 + 3.09i)14-s + (−3.23 − 2.13i)15-s − 4·16-s + (3 − 3i)17-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (0.866 + 0.499i)3-s + i·4-s + (−0.998 − 0.0599i)5-s + (−0.258 − 0.965i)6-s + (0.428 − 1.59i)7-s + (0.707 − 0.707i)8-s + (0.5 + 0.866i)9-s + (0.663 + 0.748i)10-s + (−0.301 − 0.522i)11-s + (−0.499 + 0.866i)12-s + (0.480 − 0.128i)13-s + (−1.43 + 0.827i)14-s + (−0.834 − 0.550i)15-s − 16-s + (0.727 − 0.727i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.940235 - 0.619000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.940235 - 0.619000i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 + (2.23 + 0.133i)T \) |
good | 7 | \( 1 + (-1.13 + 4.23i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.73 + 0.464i)T + (11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-3 + 3i)T - 17iT^{2} \) |
| 19 | \( 1 - 4.73T + 19T^{2} \) |
| 23 | \( 1 + (-0.866 + 0.232i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (2.76 - 1.59i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.09 + 7.09i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.464 + 0.464i)T - 37iT^{2} \) |
| 41 | \( 1 + (5.59 + 3.23i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.36 - 8.83i)T + (-37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (1.5 + 0.401i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-5.19 + 5.19i)T - 53iT^{2} \) |
| 59 | \( 1 + (1.56 + 0.901i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (12.6 - 7.33i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.96 + 7.33i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 + (10.1 - 10.1i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.90 + 1.09i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.3 - 3.03i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 - 2.66T + 89T^{2} \) |
| 97 | \( 1 + (-1.83 + 6.83i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.09959315274314655131924385448, −10.36048992341193527185769779396, −9.550714591943045044047783329498, −8.413736471502459291830082599851, −7.73251914148829699345693159581, −7.23550534001813438367942115322, −4.77358111035545628538403890067, −3.80330049668994363816354288423, −3.10132213518760162573479705627, −1.01682826259216223595647889604,
1.65907284822874193602668733296, 3.16245773785974763026513044294, 4.87294444910155861493253679899, 6.04452704003901409032919803402, 7.21644743711787199819319418150, 8.002436547017668884173336226007, 8.605720193546052719333704792865, 9.329725633796890131710415471844, 10.49504056784097918878083036057, 11.82096889609246656827386016068