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
−11.82096889609246656827386016068, −10.49504056784097918878083036057, −9.329725633796890131710415471844, −8.605720193546052719333704792865, −8.002436547017668884173336226007, −7.21644743711787199819319418150, −6.04452704003901409032919803402, −4.87294444910155861493253679899, −3.16245773785974763026513044294, −1.65907284822874193602668733296,
1.01682826259216223595647889604, 3.10132213518760162573479705627, 3.80330049668994363816354288423, 4.77358111035545628538403890067, 7.23550534001813438367942115322, 7.73251914148829699345693159581, 8.413736471502459291830082599851, 9.550714591943045044047783329498, 10.36048992341193527185769779396, 11.09959315274314655131924385448