L(s) = 1 | + (−0.909 − 0.415i)2-s + (0.989 − 0.142i)3-s + (0.654 + 0.755i)4-s + (0.281 − 0.959i)5-s + (−0.959 − 0.281i)6-s + (−0.281 − 0.959i)8-s + (0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (0.755 + 0.654i)12-s + (0.142 − 0.989i)15-s + (−0.142 + 0.989i)16-s + (0.755 + 0.345i)17-s + (−0.989 − 0.142i)18-s + (0.544 + 1.19i)19-s + (0.909 − 0.415i)20-s + ⋯ |
L(s) = 1 | + (−0.909 − 0.415i)2-s + (0.989 − 0.142i)3-s + (0.654 + 0.755i)4-s + (0.281 − 0.959i)5-s + (−0.959 − 0.281i)6-s + (−0.281 − 0.959i)8-s + (0.959 − 0.281i)9-s + (−0.654 + 0.755i)10-s + (0.755 + 0.654i)12-s + (0.142 − 0.989i)15-s + (−0.142 + 0.989i)16-s + (0.755 + 0.345i)17-s + (−0.989 − 0.142i)18-s + (0.544 + 1.19i)19-s + (0.909 − 0.415i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.094876353\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.094876353\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.909 + 0.415i)T \) |
| 3 | \( 1 + (-0.989 + 0.142i)T \) |
| 5 | \( 1 + (-0.281 + 0.959i)T \) |
| 23 | \( 1 + (0.540 + 0.841i)T \) |
good | 7 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 11 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 13 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 17 | \( 1 + (-0.755 - 0.345i)T + (0.654 + 0.755i)T^{2} \) |
| 19 | \( 1 + (-0.544 - 1.19i)T + (-0.654 + 0.755i)T^{2} \) |
| 29 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 31 | \( 1 + (1.49 + 0.215i)T + (0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 47 | \( 1 - 1.68iT - T^{2} \) |
| 53 | \( 1 + (1.27 + 1.10i)T + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (-1.07 - 0.153i)T + (0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 71 | \( 1 + (0.415 + 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 79 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 83 | \( 1 + (1.89 - 0.557i)T + (0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 97 | \( 1 + (0.841 + 0.540i)T^{2} \) |
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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
−9.709191917507704670646839481106, −8.820306149770767779494053118892, −8.136385270760057003387222726026, −7.71929068344784168372438577609, −6.61436241449413544132553753149, −5.54998802431527230932406291483, −4.16778717378194545091749358538, −3.39841791108444736633903466391, −2.14634456999094865549772318571, −1.29469880501863895370126388323,
1.67306980544804906071679410294, 2.70938856713350638943366677101, 3.53977980287752329705749544990, 5.05795985876628342747997022853, 6.00751042084721272181944240721, 7.16746178611581396156278827895, 7.35797785394743870985254234021, 8.296770604451739956456670958463, 9.289467039128050846155749035142, 9.615323804919851163402507423058