L(s) = 1 | + (−0.415 + 0.909i)2-s + (−1.94 + 0.139i)3-s + (−0.654 − 0.755i)4-s + (0.682 − 1.83i)6-s + (0.959 − 0.281i)8-s + (2.79 − 0.401i)9-s + (−0.540 − 0.158i)11-s + (1.38 + 1.38i)12-s + (−0.142 + 0.989i)16-s + (−0.755 + 0.345i)17-s + (−0.794 + 2.70i)18-s + (1.40 + 1.05i)19-s + (0.368 − 0.425i)22-s + (−1.83 + 0.682i)24-s + (−0.841 + 0.540i)25-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.909i)2-s + (−1.94 + 0.139i)3-s + (−0.654 − 0.755i)4-s + (0.682 − 1.83i)6-s + (0.959 − 0.281i)8-s + (2.79 − 0.401i)9-s + (−0.540 − 0.158i)11-s + (1.38 + 1.38i)12-s + (−0.142 + 0.989i)16-s + (−0.755 + 0.345i)17-s + (−0.794 + 2.70i)18-s + (1.40 + 1.05i)19-s + (0.368 − 0.425i)22-s + (−1.83 + 0.682i)24-s + (−0.841 + 0.540i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3262207565\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3262207565\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 89 | \( 1 + (0.841 + 0.540i)T \) |
good | 3 | \( 1 + (1.94 - 0.139i)T + (0.989 - 0.142i)T^{2} \) |
| 5 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 7 | \( 1 + (-0.540 - 0.841i)T^{2} \) |
| 11 | \( 1 + (0.540 + 0.158i)T + (0.841 + 0.540i)T^{2} \) |
| 13 | \( 1 + (0.989 - 0.142i)T^{2} \) |
| 17 | \( 1 + (0.755 - 0.345i)T + (0.654 - 0.755i)T^{2} \) |
| 19 | \( 1 + (-1.40 - 1.05i)T + (0.281 + 0.959i)T^{2} \) |
| 23 | \( 1 + (0.281 + 0.959i)T^{2} \) |
| 29 | \( 1 + (0.540 + 0.841i)T^{2} \) |
| 31 | \( 1 + (0.281 - 0.959i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (0.100 - 1.41i)T + (-0.989 - 0.142i)T^{2} \) |
| 43 | \( 1 + (-0.898 - 1.64i)T + (-0.540 + 0.841i)T^{2} \) |
| 47 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 53 | \( 1 + (-0.142 - 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.697 - 0.0498i)T + (0.989 + 0.142i)T^{2} \) |
| 61 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 67 | \( 1 + (-1.29 + 1.49i)T + (-0.142 - 0.989i)T^{2} \) |
| 71 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 73 | \( 1 + (0.153 - 1.07i)T + (-0.959 - 0.281i)T^{2} \) |
| 79 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 83 | \( 1 + (0.697 - 1.86i)T + (-0.755 - 0.654i)T^{2} \) |
| 97 | \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \) |
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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
−10.95689352607128272411107005354, −9.962773069106447534756336618066, −9.551603650082827202893110056990, −8.022179239277086245051934346768, −7.29028515215613751174928893186, −6.32295256261687292957502672261, −5.72817552213318340738432457170, −5.00716593730445465197913814563, −4.04343792951888424511216687459, −1.28094185708622593531553461456,
0.60152963256656413118709655715, 2.19534363424892827770481020801, 3.96681418739508165666320035133, 4.95471288819158337197923390615, 5.60868631443069227309641886621, 6.97533845391762948453589368613, 7.48484938299490647869404943563, 8.918587054406445712368464916976, 9.931033158754207989073953103991, 10.46734482867067852083148443855