L(s) = 1 | − i·2-s + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s − i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s − 13-s + (−0.866 + 0.5i)14-s + (−0.499 − 0.866i)15-s − 16-s + i·17-s + (0.866 − 0.499i)18-s + 0.999i·21-s + ⋯ |
L(s) = 1 | − i·2-s + (−0.866 − 0.5i)3-s + (0.866 + 0.5i)5-s + (−0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s − i·8-s + (0.499 + 0.866i)9-s + (0.5 − 0.866i)10-s − 13-s + (−0.866 + 0.5i)14-s + (−0.499 − 0.866i)15-s − 16-s + i·17-s + (0.866 − 0.499i)18-s + 0.999i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.190 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6943622040\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6943622040\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 + iT - T^{2} \) |
| 5 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - iT - T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - iT - T^{2} \) |
| 29 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + iT - T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + iT - T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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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
−11.81635092291064623888707169402, −10.88761874408485011827668634404, −10.16590117329904319655039105015, −9.739224475355644980804424341740, −7.71822027925855096827287524729, −6.72194297379546005470977822708, −6.08316196652020099419979268741, −4.49079745162150097735768487124, −2.92245919067083545283945155759, −1.56870668635183199225822516103,
2.52377090036080620067152435299, 4.79555038552527192406901869376, 5.48172561341015488566674130080, 6.29154362102306211253296752947, 7.16237759234306319121707143661, 8.657188688802703417166928955622, 9.500996823775302586012334606579, 10.31687567105922283440639059795, 11.65151918990563666516021581874, 12.22814614557714800690323182148