L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)11-s + 0.999i·15-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.866 + 0.5i)23-s − i·27-s + (0.866 − 0.5i)31-s + (0.499 − 0.866i)33-s + (0.5 − 0.866i)37-s + (0.866 + 0.5i)47-s + (−0.866 − 0.499i)51-s + (0.5 + 0.866i)53-s + 0.999i·55-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)3-s + (0.5 − 0.866i)5-s + (−0.866 + 0.5i)11-s + 0.999i·15-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (0.866 + 0.5i)23-s − i·27-s + (0.866 − 0.5i)31-s + (0.499 − 0.866i)33-s + (0.5 − 0.866i)37-s + (0.866 + 0.5i)47-s + (−0.866 − 0.499i)51-s + (0.5 + 0.866i)53-s + 0.999i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8676816006\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8676816006\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−9.758689730363776328757178994256, −9.115350708849319105978166118770, −8.070495321821282805589169263859, −7.43165878805129305196201446815, −6.09424971939573400825935975273, −5.50093497655651430387291733205, −4.96068044842285934995907540495, −4.05779209180418838653092313255, −2.65704087481342035171870475502, −1.26464359384722805519026021915,
0.909574632935171645474804838652, 2.60913413547070431681398585092, 3.22859260101081769004929314966, 4.90738215615066665557081244035, 5.49871722901946486139459661329, 6.40967540526245773432452434970, 6.91706419275973185287952284097, 7.73427428282035611713744636825, 8.757618693543907015833308156371, 9.714133528901200975852726530392