L(s) = 1 | + (0.891 + 0.453i)2-s + (0.309 + 0.951i)3-s + (0.587 + 0.809i)4-s + (−0.156 + 0.987i)6-s + (0.156 + 0.987i)8-s + (−0.809 + 0.587i)9-s + (−1.65 − 0.398i)11-s + (−0.587 + 0.809i)12-s + (−0.309 + 0.951i)16-s + (0.652 − 0.399i)17-s + (−0.987 + 0.156i)18-s + (0.987 − 1.15i)19-s + (−1.29 − 1.10i)22-s + (−0.891 + 0.453i)24-s + (0.951 + 0.309i)25-s + ⋯ |
L(s) = 1 | + (0.891 + 0.453i)2-s + (0.309 + 0.951i)3-s + (0.587 + 0.809i)4-s + (−0.156 + 0.987i)6-s + (0.156 + 0.987i)8-s + (−0.809 + 0.587i)9-s + (−1.65 − 0.398i)11-s + (−0.587 + 0.809i)12-s + (−0.309 + 0.951i)16-s + (0.652 − 0.399i)17-s + (−0.987 + 0.156i)18-s + (0.987 − 1.15i)19-s + (−1.29 − 1.10i)22-s + (−0.891 + 0.453i)24-s + (0.951 + 0.309i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.721468810\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.721468810\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.891 - 0.453i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 41 | \( 1 + (-0.987 - 0.156i)T \) |
good | 5 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 7 | \( 1 + (0.987 - 0.156i)T^{2} \) |
| 11 | \( 1 + (1.65 + 0.398i)T + (0.891 + 0.453i)T^{2} \) |
| 13 | \( 1 + (0.156 - 0.987i)T^{2} \) |
| 17 | \( 1 + (-0.652 + 0.399i)T + (0.453 - 0.891i)T^{2} \) |
| 19 | \( 1 + (-0.987 + 1.15i)T + (-0.156 - 0.987i)T^{2} \) |
| 23 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.453 - 0.891i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (-0.863 + 1.69i)T + (-0.587 - 0.809i)T^{2} \) |
| 47 | \( 1 + (-0.987 - 0.156i)T^{2} \) |
| 53 | \( 1 + (0.453 + 0.891i)T^{2} \) |
| 59 | \( 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 67 | \( 1 + (1.89 - 0.453i)T + (0.891 - 0.453i)T^{2} \) |
| 71 | \( 1 + (0.891 + 0.453i)T^{2} \) |
| 73 | \( 1 + (-0.437 - 0.437i)T + iT^{2} \) |
| 79 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 83 | \( 1 - 1.61iT - T^{2} \) |
| 89 | \( 1 + (0.465 - 0.0366i)T + (0.987 - 0.156i)T^{2} \) |
| 97 | \( 1 + (0.303 + 1.26i)T + (-0.891 + 0.453i)T^{2} \) |
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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
−10.68085871598502744071630237065, −9.566116324755092258711796193005, −8.682145438585542714369912161269, −7.81926301532307849311679437843, −7.15954380516200977882753397698, −5.71182814211857947400518741103, −5.22910781139128738852869469377, −4.43137764451061135545004630581, −3.14247993770594581166578155721, −2.69135851373024059624198269353,
1.40547329776343704825756767938, 2.61882927134268631539943010587, 3.33609651031297984890755096906, 4.74719333734914672631796684197, 5.65530046043200760569957273055, 6.35293825387112036680442165216, 7.63229704223319810427252280802, 7.82125308395911181657874812515, 9.259241436958217216952236429543, 10.18059888065604997901129431293