L(s) = 1 | + (0.540 − 0.841i)2-s + (0.755 − 0.654i)3-s + (−0.415 − 0.909i)4-s + (0.989 − 0.142i)5-s + (−0.142 − 0.989i)6-s + (−0.989 − 0.142i)8-s + (0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (−0.909 − 0.415i)12-s + (0.654 − 0.755i)15-s + (−0.654 + 0.755i)16-s + (−0.909 + 1.41i)17-s + (−0.755 − 0.654i)18-s + (−0.698 + 0.449i)19-s + (−0.540 − 0.841i)20-s + ⋯ |
L(s) = 1 | + (0.540 − 0.841i)2-s + (0.755 − 0.654i)3-s + (−0.415 − 0.909i)4-s + (0.989 − 0.142i)5-s + (−0.142 − 0.989i)6-s + (−0.989 − 0.142i)8-s + (0.142 − 0.989i)9-s + (0.415 − 0.909i)10-s + (−0.909 − 0.415i)12-s + (0.654 − 0.755i)15-s + (−0.654 + 0.755i)16-s + (−0.909 + 1.41i)17-s + (−0.755 − 0.654i)18-s + (−0.698 + 0.449i)19-s + (−0.540 − 0.841i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.911601623\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.911601623\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.540 + 0.841i)T \) |
| 3 | \( 1 + (-0.755 + 0.654i)T \) |
| 5 | \( 1 + (-0.989 + 0.142i)T \) |
| 23 | \( 1 + (0.281 - 0.959i)T \) |
good | 7 | \( 1 + (0.654 + 0.755i)T^{2} \) |
| 11 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 13 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 17 | \( 1 + (0.909 - 1.41i)T + (-0.415 - 0.909i)T^{2} \) |
| 19 | \( 1 + (0.698 - 0.449i)T + (0.415 - 0.909i)T^{2} \) |
| 29 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 31 | \( 1 + (-1.37 - 1.19i)T + (0.142 + 0.989i)T^{2} \) |
| 37 | \( 1 + (-0.959 - 0.281i)T^{2} \) |
| 41 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 43 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 47 | \( 1 + 1.91iT - T^{2} \) |
| 53 | \( 1 + (1.74 + 0.797i)T + (0.654 + 0.755i)T^{2} \) |
| 59 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 61 | \( 1 + (-0.425 - 0.368i)T + (0.142 + 0.989i)T^{2} \) |
| 67 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 71 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 73 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
| 79 | \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \) |
| 83 | \( 1 + (0.215 - 1.49i)T + (-0.959 - 0.281i)T^{2} \) |
| 89 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 97 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
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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
−9.692442845928838812335854691631, −8.694327968753511490336799190423, −8.298914073264932852764279003910, −6.72354984070560108787948112340, −6.28345292488409834428619372028, −5.29598879956987603002808789614, −4.16880888113910354756801206126, −3.24115141382537317918521450525, −2.12591924375118151141751664417, −1.52900475071445456830066682284,
2.40143048139237154315328407795, 2.97691912364608196714954239650, 4.49861186488070511873580481759, 4.72256981848983171044082677677, 5.98803753806924201330981606515, 6.63110097451866948695452360833, 7.61833860735576161732749769217, 8.432975981674939287835580248493, 9.256407982210935430855627061947, 9.625175606863439935620033095863