L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s − i·5-s + (0.866 − 0.499i)6-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s − 0.999·12-s − 13-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + 0.999i·18-s + (0.866 − 0.499i)20-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s − i·5-s + (0.866 − 0.499i)6-s − 0.999i·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (−0.866 − 0.5i)11-s − 0.999·12-s − 13-s + (0.866 + 0.5i)15-s + (−0.5 + 0.866i)16-s + 0.999i·18-s + (0.866 − 0.499i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 - 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.06717392729\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.06717392729\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 - 1.73iT - T^{2} \) |
| 37 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + iT - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
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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
−10.00990441910185301538492168799, −9.160355145380193299860613608221, −8.735912928508348893925137844610, −7.78587432591920244316233945023, −6.96847999741343284157400054131, −5.57110298786454059128633441288, −5.12475843274100895489732061141, −3.94108365881081078480346490966, −3.10181688433294231286970377255, −1.63352120373808205776458497356,
0.06972353008464826317171877562, 2.11834542637602205600736471499, 2.55972255514505600877049159427, 4.50929015634977972270348151579, 5.62081852328069182155219350527, 6.24411238120174680504624130616, 7.02352342515995336653169141074, 7.71206520858472841767538415089, 8.049131326399447852394282689838, 9.359504145562308865929698797132