L(s) = 1 | + (−0.623 + 1.07i)3-s + (−0.5 − 0.866i)4-s + 0.445·5-s + (−0.277 − 0.480i)9-s + (0.5 − 0.866i)11-s + 1.24·12-s + (−0.277 + 0.480i)15-s + (−0.499 + 0.866i)16-s + (−0.222 − 0.385i)20-s + (0.900 − 1.56i)23-s − 0.801·25-s − 0.554·27-s + 1.80·31-s + (0.623 + 1.07i)33-s + (−0.277 + 0.480i)36-s + (0.623 − 1.07i)37-s + ⋯ |
L(s) = 1 | + (−0.623 + 1.07i)3-s + (−0.5 − 0.866i)4-s + 0.445·5-s + (−0.277 − 0.480i)9-s + (0.5 − 0.866i)11-s + 1.24·12-s + (−0.277 + 0.480i)15-s + (−0.499 + 0.866i)16-s + (−0.222 − 0.385i)20-s + (0.900 − 1.56i)23-s − 0.801·25-s − 0.554·27-s + 1.80·31-s + (0.623 + 1.07i)33-s + (−0.277 + 0.480i)36-s + (0.623 − 1.07i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9063723651\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9063723651\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 3 | \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 - 0.445T + T^{2} \) |
| 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 | \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 - 1.80T + T^{2} \) |
| 37 | \( 1 + (-0.623 + 1.07i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 - 1.80T + T^{2} \) |
| 53 | \( 1 + 0.445T + T^{2} \) |
| 59 | \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 1.07i)T + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.900 - 1.56i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (0.222 + 0.385i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−9.508419019715048210265770449417, −8.973014509282867355699069972971, −8.097823038966850181505804740065, −6.66084026066184654081025110937, −6.00187454203816990084579354249, −5.39521957652060785700901466208, −4.54513207177709199599726641901, −3.96021448402681239518612304736, −2.51502387273918812684842421364, −0.910403748579808583283472202124,
1.20862475677991773526532489167, 2.36536180799443466949155882835, 3.58248623062528508787844967306, 4.58326091166527361973185677157, 5.49303302276095066179593624009, 6.43698357121179103462054350193, 7.09237698044942007025878474989, 7.70830576893491226952158419947, 8.513338665063175819115351140374, 9.534889709872140009227953596808