L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (−0.866 − 0.499i)6-s + (0.633 − 1.09i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−1.09 + 0.633i)11-s − 0.999i·12-s + (3.23 + 1.59i)13-s + 1.26·14-s + (−0.5 − 0.866i)16-s + (−4.5 − 2.59i)17-s + 0.999·18-s + (4.09 + 2.36i)19-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.353 − 0.204i)6-s + (0.239 − 0.415i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.331 + 0.191i)11-s − 0.288i·12-s + (0.896 + 0.443i)13-s + 0.338·14-s + (−0.125 − 0.216i)16-s + (−1.09 − 0.630i)17-s + 0.235·18-s + (0.940 + 0.542i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00837 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00837 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.728476595\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.728476595\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.23 - 1.59i)T \) |
good | 7 | \( 1 + (-0.633 + 1.09i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.09 - 0.633i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (4.5 + 2.59i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.09 - 2.36i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.09 + 4.09i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 + 2.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 9.46iT - 31T^{2} \) |
| 37 | \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.59 + 3.23i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.63 + 2.09i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + (-12 - 6.92i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.59 - 13.1i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (3.63 + 6.29i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.90 - 1.09i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 + 8.39T + 79T^{2} \) |
| 83 | \( 1 + 5.66T + 83T^{2} \) |
| 89 | \( 1 + (-8.19 + 4.73i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3 - 5.19i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.093975712191692578297101110246, −8.737473507879877908600591703501, −7.55160894855942862707553501039, −6.97808705743515482778261963456, −6.22605211320916287812098274568, −5.29685853754478564160501263475, −4.63498196904425243087019416193, −3.85242805544941582113810774158, −2.72855789663734331184171696247, −1.05688471777580820580716598283,
0.75083650722170635970574602903, 1.94507736298041121853131279670, 3.00102504570790996074387821184, 3.98723087260874925067378693124, 5.05333550475732878482386521548, 5.62550279266785482711479197376, 6.43464262524988442818737724412, 7.38031285456726670229852904857, 8.294436978191395994260510287000, 9.087568549960199432936103855144