L(s) = 1 | + (0.826 + 0.563i)2-s + (−1.07 − 0.997i)3-s + (0.365 + 0.930i)4-s + (−0.326 − 1.42i)6-s + (0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.0858 + 1.14i)9-s + (−0.123 + 1.64i)11-s + (0.535 − 1.36i)12-s + (−1.48 + 0.716i)13-s + (0.733 − 0.680i)14-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (−0.574 + 0.995i)18-s + (−1.21 + 0.825i)21-s + (−1.03 + 1.29i)22-s + ⋯ |
L(s) = 1 | + (0.826 + 0.563i)2-s + (−1.07 − 0.997i)3-s + (0.365 + 0.930i)4-s + (−0.326 − 1.42i)6-s + (0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.0858 + 1.14i)9-s + (−0.123 + 1.64i)11-s + (0.535 − 1.36i)12-s + (−1.48 + 0.716i)13-s + (0.733 − 0.680i)14-s + (−0.733 + 0.680i)16-s + (−0.988 + 0.149i)17-s + (−0.574 + 0.995i)18-s + (−1.21 + 0.825i)21-s + (−1.03 + 1.29i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5655712303\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5655712303\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.826 - 0.563i)T \) |
| 7 | \( 1 + (-0.222 + 0.974i)T \) |
| 17 | \( 1 + (0.988 - 0.149i)T \) |
good | 3 | \( 1 + (1.07 + 0.997i)T + (0.0747 + 0.997i)T^{2} \) |
| 5 | \( 1 + (-0.826 + 0.563i)T^{2} \) |
| 11 | \( 1 + (0.123 - 1.64i)T + (-0.988 - 0.149i)T^{2} \) |
| 13 | \( 1 + (1.48 - 0.716i)T + (0.623 - 0.781i)T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (1.78 + 0.268i)T + (0.955 + 0.294i)T^{2} \) |
| 29 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 31 | \( 1 + (0.222 - 0.385i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 41 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 43 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 47 | \( 1 + (-0.365 - 0.930i)T^{2} \) |
| 53 | \( 1 + (-0.698 - 1.77i)T + (-0.733 + 0.680i)T^{2} \) |
| 59 | \( 1 + (-0.826 - 0.563i)T^{2} \) |
| 61 | \( 1 + (0.733 + 0.680i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (1.19 - 1.49i)T + (-0.222 - 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.365 + 0.930i)T^{2} \) |
| 79 | \( 1 + (-0.0747 - 0.129i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (0.109 + 1.46i)T + (-0.988 + 0.149i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.909293364690997518721247968169, −7.73505207635000942397959219361, −7.36024355992665126890247456768, −6.82737433690496209991234595145, −6.32565259254648837620293637717, −5.29375752180686169408571806546, −4.49614917654524318301186402147, −4.23123927947657641857296758353, −2.46079896182330565401828073094, −1.76624598299137482936908931135,
0.26391246481764914994674035999, 2.13777570569629727145661093096, 3.02607811388920478406258425586, 3.92713418902667076649982375014, 4.86746797975568047600902769535, 5.32921170082770696440677351730, 5.87941446264543156158893104242, 6.51159209323618226020582098493, 7.75964939134610132209970347450, 8.764420802922868595570633665084