L(s) = 1 | + (0.654 − 0.755i)2-s + (0.559 − 0.418i)3-s + (−0.142 − 0.989i)4-s + (0.0498 − 0.697i)6-s + (−0.841 − 0.540i)8-s + (−0.144 + 0.490i)9-s + (0.909 − 0.584i)11-s + (−0.494 − 0.494i)12-s + (−0.959 + 0.281i)16-s + (−0.989 + 0.857i)17-s + (0.276 + 0.430i)18-s + (−0.373 + 0.203i)19-s + (0.153 − 1.07i)22-s + (−0.697 + 0.0498i)24-s + (−0.415 − 0.909i)25-s + ⋯ |
L(s) = 1 | + (0.654 − 0.755i)2-s + (0.559 − 0.418i)3-s + (−0.142 − 0.989i)4-s + (0.0498 − 0.697i)6-s + (−0.841 − 0.540i)8-s + (−0.144 + 0.490i)9-s + (0.909 − 0.584i)11-s + (−0.494 − 0.494i)12-s + (−0.959 + 0.281i)16-s + (−0.989 + 0.857i)17-s + (0.276 + 0.430i)18-s + (−0.373 + 0.203i)19-s + (0.153 − 1.07i)22-s + (−0.697 + 0.0498i)24-s + (−0.415 − 0.909i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0176 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0176 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.478354529\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.478354529\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 89 | \( 1 + (0.415 - 0.909i)T \) |
good | 3 | \( 1 + (-0.559 + 0.418i)T + (0.281 - 0.959i)T^{2} \) |
| 5 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 7 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 11 | \( 1 + (-0.909 + 0.584i)T + (0.415 - 0.909i)T^{2} \) |
| 13 | \( 1 + (0.281 - 0.959i)T^{2} \) |
| 17 | \( 1 + (0.989 - 0.857i)T + (0.142 - 0.989i)T^{2} \) |
| 19 | \( 1 + (0.373 - 0.203i)T + (0.540 - 0.841i)T^{2} \) |
| 23 | \( 1 + (0.540 - 0.841i)T^{2} \) |
| 29 | \( 1 + (-0.909 + 0.415i)T^{2} \) |
| 31 | \( 1 + (0.540 + 0.841i)T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (0.847 - 1.13i)T + (-0.281 - 0.959i)T^{2} \) |
| 43 | \( 1 + (-1.94 - 0.424i)T + (0.909 + 0.415i)T^{2} \) |
| 47 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 53 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 59 | \( 1 + (0.114 + 0.0855i)T + (0.281 + 0.959i)T^{2} \) |
| 61 | \( 1 + (-0.755 + 0.654i)T^{2} \) |
| 67 | \( 1 + (-0.0801 + 0.557i)T + (-0.959 - 0.281i)T^{2} \) |
| 71 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 73 | \( 1 + (-1.74 + 0.512i)T + (0.841 - 0.540i)T^{2} \) |
| 79 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 83 | \( 1 + (-0.114 + 1.59i)T + (-0.989 - 0.142i)T^{2} \) |
| 97 | \( 1 + (1.61 + 1.03i)T + (0.415 + 0.909i)T^{2} \) |
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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
−10.71359201733046520096223498425, −9.603582127285937841876291084370, −8.760277816752352989403393939973, −8.009936558423526325644290904123, −6.63055941871353361677202439105, −5.99297532541099505225463794933, −4.69112155324992936624992568172, −3.79522274678459577490522500100, −2.64240277343721040094180467435, −1.61745954927173754872982462118,
2.44044882206428685175121067365, 3.67033557484111286334084862494, 4.33309698698171464359716986826, 5.42180083046326640442121700770, 6.57858593326409785509955167941, 7.13210131258764936278332439312, 8.288287112904915972025449126071, 9.146505201597515644450778308280, 9.526933714519028639902868445474, 11.00233496114232303031111811842