L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + 5-s + (−0.499 − 0.866i)9-s − 1.73i·11-s − 0.999·12-s + (1.5 + 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.499 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)20-s + 25-s + 0.999·27-s + (1.49 + 0.866i)33-s + (0.499 − 0.866i)36-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + 5-s + (−0.499 − 0.866i)9-s − 1.73i·11-s − 0.999·12-s + (1.5 + 0.866i)13-s + (−0.5 + 0.866i)15-s + (−0.499 + 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 + 0.866i)20-s + 25-s + 0.999·27-s + (1.49 + 0.866i)33-s + (0.499 − 0.866i)36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.381076063\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.381076063\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + 1.73iT - T^{2} \) |
| 13 | \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 1.73iT - T^{2} \) |
| 73 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
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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
−9.179291040763371791126752603730, −8.775422706104966744135415091549, −8.161810151030676458203986576107, −6.70782461752361003388159657240, −6.17050651741043198359451326295, −5.75066415523741377470466000794, −4.44865694576175685041412409581, −3.63234749344303478632032259890, −2.94312829310744692393409048619, −1.53415641740609884891509966318,
1.17557768724919395806095352179, 1.95545074615478379152446187518, 2.78930609763453276351320689692, 4.56619330363123953541590790927, 5.41827720841477317523960941954, 5.89895402691265616349141089682, 6.82891119589023998683851487640, 7.08843877231934599915486727115, 8.246689150730527971556058386911, 9.187286126930038455213650910060