L(s) = 1 | + (−0.993 − 0.116i)2-s + (−0.0581 + 0.998i)3-s + (0.973 + 0.230i)4-s + (0.893 − 0.448i)5-s + (0.173 − 0.984i)6-s + (0.479 − 1.60i)7-s + (−0.939 − 0.342i)8-s + (−0.993 − 0.116i)9-s + (−0.939 + 0.342i)10-s + (−0.286 + 0.957i)12-s + (−0.661 + 1.53i)14-s + (0.396 + 0.918i)15-s + (0.893 + 0.448i)16-s + (0.973 + 0.230i)18-s + (0.973 − 0.230i)20-s + (1.57 + 0.571i)21-s + ⋯ |
L(s) = 1 | + (−0.993 − 0.116i)2-s + (−0.0581 + 0.998i)3-s + (0.973 + 0.230i)4-s + (0.893 − 0.448i)5-s + (0.173 − 0.984i)6-s + (0.479 − 1.60i)7-s + (−0.939 − 0.342i)8-s + (−0.993 − 0.116i)9-s + (−0.939 + 0.342i)10-s + (−0.286 + 0.957i)12-s + (−0.661 + 1.53i)14-s + (0.396 + 0.918i)15-s + (0.893 + 0.448i)16-s + (0.973 + 0.230i)18-s + (0.973 − 0.230i)20-s + (1.57 + 0.571i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 + 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8387719167\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8387719167\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.993 + 0.116i)T \) |
| 3 | \( 1 + (0.0581 - 0.998i)T \) |
| 5 | \( 1 + (-0.893 + 0.448i)T \) |
good | 7 | \( 1 + (-0.479 + 1.60i)T + (-0.835 - 0.549i)T^{2} \) |
| 11 | \( 1 + (0.993 - 0.116i)T^{2} \) |
| 13 | \( 1 + (0.286 - 0.957i)T^{2} \) |
| 17 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 19 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.164 - 0.549i)T + (-0.835 + 0.549i)T^{2} \) |
| 29 | \( 1 + (0.543 + 1.26i)T + (-0.686 + 0.727i)T^{2} \) |
| 31 | \( 1 + (0.0581 + 0.998i)T^{2} \) |
| 37 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 41 | \( 1 + (1.18 - 0.138i)T + (0.973 - 0.230i)T^{2} \) |
| 43 | \( 1 + (-1.57 + 1.03i)T + (0.396 - 0.918i)T^{2} \) |
| 47 | \( 1 + (-1.36 - 1.44i)T + (-0.0581 + 0.998i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.993 + 0.116i)T^{2} \) |
| 61 | \( 1 + (0.113 - 0.0268i)T + (0.893 - 0.448i)T^{2} \) |
| 67 | \( 1 + (-0.313 + 0.727i)T + (-0.686 - 0.727i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.973 - 0.230i)T^{2} \) |
| 83 | \( 1 + (1.18 + 0.138i)T + (0.973 + 0.230i)T^{2} \) |
| 89 | \( 1 + (-1.86 - 0.679i)T + (0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.597 - 0.802i)T^{2} \) |
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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
−9.549663450636930732467577529740, −9.034490373631036596082568588706, −8.066084132928010816223298114536, −7.38770941590066498154272597219, −6.34054731639104959735051972558, −5.49857881866288711541163620112, −4.44526357030582563300758680270, −3.61680962675085140229396312470, −2.30017651302046543405861657908, −0.976965450279633691849217498570,
1.52888332527984293629154593507, 2.30447945362343786712977226103, 2.97750638966527413938790887945, 5.32226192193410047024196593637, 5.73087478101002449731547211656, 6.56323847494876619928041096037, 7.22322072109484084811417718402, 8.190657214782065081332517978849, 8.866897027996998248081553287069, 9.288515830420581007252810186196