L(s) = 1 | + (−1.59 − 1.16i)2-s + (−0.309 + 0.951i)3-s + (0.896 + 2.76i)4-s + (1.59 − 1.16i)6-s + (1.16 − 3.57i)8-s + (−0.809 − 0.587i)9-s + (−0.453 + 0.891i)11-s − 2.90·12-s + (0.951 + 0.690i)13-s + (−3.65 + 2.65i)16-s + (0.734 − 0.533i)17-s + (0.610 + 1.87i)18-s + (0.587 − 1.80i)19-s + (1.76 − 0.896i)22-s + 0.312·23-s + (3.03 + 2.20i)24-s + ⋯ |
L(s) = 1 | + (−1.59 − 1.16i)2-s + (−0.309 + 0.951i)3-s + (0.896 + 2.76i)4-s + (1.59 − 1.16i)6-s + (1.16 − 3.57i)8-s + (−0.809 − 0.587i)9-s + (−0.453 + 0.891i)11-s − 2.90·12-s + (0.951 + 0.690i)13-s + (−3.65 + 2.65i)16-s + (0.734 − 0.533i)17-s + (0.610 + 1.87i)18-s + (0.587 − 1.80i)19-s + (1.76 − 0.896i)22-s + 0.312·23-s + (3.03 + 2.20i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4852535592\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4852535592\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (0.453 - 0.891i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
good | 2 | \( 1 + (1.59 + 1.16i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 17 | \( 1 + (-0.734 + 0.533i)T + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.587 + 1.80i)T + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 - 0.312T + T^{2} \) |
| 29 | \( 1 + (-0.0966 - 0.297i)T + (-0.809 + 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (-1.59 - 1.16i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.550 + 1.69i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (1.14 - 0.831i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 - 0.907T + T^{2} \) |
| 97 | \( 1 + (0.309 + 0.951i)T^{2} \) |
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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
−9.361545522635880362145810526402, −8.970135157740281171812690987934, −8.158350349838644512508943122793, −7.21557220939545658635586182526, −6.53582368575083875791290771893, −4.98768370274604267830291410010, −4.15131215957184382979810895111, −3.16574436270994045511738618163, −2.41462667961955188958378324491, −0.929316804963404035556181063398,
0.908207317474104170195334532246, 1.73252143870362796964371001492, 3.23989939922879071389592503232, 5.33467587050083111257618190156, 5.76352124991637515693168357185, 6.27135349087650206820881067608, 7.28974762968105947401783170338, 7.86164762769791967031086848557, 8.352248882090240702450501967735, 8.950332347479972230745548710203