L(s) = 1 | + (0.222 − 0.385i)3-s + (0.5 + 0.866i)4-s − 1.80i·5-s + (0.400 + 0.694i)9-s + (0.866 + 0.5i)11-s + 0.445·12-s + (−0.694 − 0.400i)15-s + (−0.499 + 0.866i)16-s + (1.56 − 0.900i)20-s + (0.623 − 1.07i)23-s − 2.24·25-s + 0.801·27-s + 1.24i·31-s + (0.385 − 0.222i)33-s + (−0.400 + 0.694i)36-s + (−0.385 − 0.222i)37-s + ⋯ |
L(s) = 1 | + (0.222 − 0.385i)3-s + (0.5 + 0.866i)4-s − 1.80i·5-s + (0.400 + 0.694i)9-s + (0.866 + 0.5i)11-s + 0.445·12-s + (−0.694 − 0.400i)15-s + (−0.499 + 0.866i)16-s + (1.56 − 0.900i)20-s + (0.623 − 1.07i)23-s − 2.24·25-s + 0.801·27-s + 1.24i·31-s + (0.385 − 0.222i)33-s + (−0.400 + 0.694i)36-s + (−0.385 − 0.222i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.487467019\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.487467019\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.866 - 0.5i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + 1.80iT - T^{2} \) |
| 7 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - 1.24iT - T^{2} \) |
| 37 | \( 1 + (0.385 + 0.222i)T + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + 1.24iT - T^{2} \) |
| 53 | \( 1 + 1.80T + T^{2} \) |
| 59 | \( 1 + (-0.385 + 0.222i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.385 - 0.222i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + (-1.07 - 0.623i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (1.56 - 0.900i)T + (0.5 - 0.866i)T^{2} \) |
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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
−8.923245345334708986724574512551, −8.684974015703302138042045219077, −7.88064721429151596640714846023, −7.15449695572697720850167118528, −6.36925914887158471323021776219, −5.03837793663575528986532904317, −4.55128081342638344216639750482, −3.57252081389335338897341488203, −2.18422802840412724414080244246, −1.36225303869284706303522324955,
1.48501976870867807202008471957, 2.79279637228783034137670805148, 3.42423741575937864993172411828, 4.41523618369482254243285754342, 5.87967771373879098467064669543, 6.25476982048119851396957850650, 7.04040991524606901398572057006, 7.59602050020837391824860133987, 9.033897380574399630040768822181, 9.647454127914674057343513166762