L(s) = 1 | + (0.587 − 0.809i)3-s + (0.587 + 0.809i)4-s + (0.951 + 0.309i)5-s + (−0.309 − 0.951i)9-s + 12-s + (0.809 − 0.587i)15-s + (−0.309 + 0.951i)16-s + (0.309 + 0.951i)20-s + (−1 − i)23-s + (0.809 + 0.587i)25-s + (−0.951 − 0.309i)27-s + (0.587 − 0.809i)36-s + (−1.39 − 0.221i)37-s − i·45-s + (1.39 − 0.221i)47-s + (0.587 + 0.809i)48-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)3-s + (0.587 + 0.809i)4-s + (0.951 + 0.309i)5-s + (−0.309 − 0.951i)9-s + 12-s + (0.809 − 0.587i)15-s + (−0.309 + 0.951i)16-s + (0.309 + 0.951i)20-s + (−1 − i)23-s + (0.809 + 0.587i)25-s + (−0.951 − 0.309i)27-s + (0.587 − 0.809i)36-s + (−1.39 − 0.221i)37-s − i·45-s + (1.39 − 0.221i)47-s + (0.587 + 0.809i)48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.788253360\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.788253360\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.587 + 0.809i)T \) |
| 5 | \( 1 + (-0.951 - 0.309i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 7 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 13 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (0.587 - 0.809i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (1 + i)T + iT^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (1.39 + 0.221i)T + (0.951 + 0.309i)T^{2} \) |
| 41 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-1.39 + 0.221i)T + (0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (0.642 - 1.26i)T + (-0.587 - 0.809i)T^{2} \) |
| 59 | \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (1 + i)T + iT^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 79 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.642 + 1.26i)T + (-0.587 - 0.809i)T^{2} \) |
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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
−9.132439968704015608647037316237, −8.692626391584907674051777744619, −7.71517174689961644107195917205, −7.17178969833146395114178261431, −6.35310951219596576847330566789, −5.80811663667059189441637376898, −4.29354903793552487199441829448, −3.20022035866513305054314161412, −2.47213697560401023017591924287, −1.66253093286661125159046525303,
1.62244923440800475987779718734, 2.40983579900300716172195148637, 3.52366881732014840383831392483, 4.71041259104091355513745251200, 5.44549072238680958288104697110, 6.04954123423968237103988564803, 7.06281275344339944165591053550, 8.014502542511690357741725639752, 8.990247853083397431982424842278, 9.501036892048083563027052913293