L(s) = 1 | + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.913 + 0.406i)5-s + (0.913 + 0.406i)9-s + (−0.5 − 0.866i)12-s + (0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (−0.913 − 0.406i)20-s + (−1 + 1.73i)23-s + (−0.809 − 0.587i)27-s + (0.104 + 0.994i)31-s + (0.309 + 0.951i)36-s + (−0.309 + 0.951i)37-s − 1.00·45-s + (−0.669 + 0.743i)47-s + (0.309 − 0.951i)48-s + ⋯ |
L(s) = 1 | + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.913 + 0.406i)5-s + (0.913 + 0.406i)9-s + (−0.5 − 0.866i)12-s + (0.978 − 0.207i)15-s + (−0.104 + 0.994i)16-s + (−0.913 − 0.406i)20-s + (−1 + 1.73i)23-s + (−0.809 − 0.587i)27-s + (0.104 + 0.994i)31-s + (0.309 + 0.951i)36-s + (−0.309 + 0.951i)37-s − 1.00·45-s + (−0.669 + 0.743i)47-s + (0.309 − 0.951i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.150 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6397905438\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6397905438\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.978 + 0.207i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 5 | \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 7 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 13 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 31 | \( 1 + (-0.104 - 0.994i)T + (-0.978 + 0.207i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \) |
| 53 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (0.669 + 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 61 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.669 - 0.743i)T^{2} \) |
| 83 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 89 | \( 1 - 2T + T^{2} \) |
| 97 | \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \) |
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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
−10.58063060792131867587539652356, −9.648308314723276264499411741686, −8.290258497566269184083671147842, −7.64577427170218415194321694083, −7.02354476220918954837824786950, −6.26477300428120776307756051371, −5.22131843092024830248341580490, −4.02619895337222724978167091349, −3.27487907789363666989478891112, −1.76944399601493162697272776785,
0.65512044014318118303178133755, 2.23857340382195697926311143151, 3.92197280252426064046130537760, 4.66070933773277285325999562066, 5.66156032869316722493400603540, 6.35403066036777922430527266735, 7.20500228806649001127495886145, 8.060708842097154743994219299699, 9.160436654833881457070489325073, 10.15333700890810975188252538641