L(s) = 1 | + i·2-s + (1.40 − 0.809i)5-s + (−0.309 + 0.535i)7-s + i·8-s + (0.809 + 1.40i)10-s + 1.61i·11-s + (0.809 − 1.40i)13-s + (−0.535 − 0.309i)14-s − 16-s + (−1.40 − 0.809i)17-s + (−0.309 − 0.535i)19-s − 1.61·22-s + (−0.535 + 0.309i)23-s + (0.809 − 1.40i)25-s + (1.40 + 0.809i)26-s + ⋯ |
L(s) = 1 | + i·2-s + (1.40 − 0.809i)5-s + (−0.309 + 0.535i)7-s + i·8-s + (0.809 + 1.40i)10-s + 1.61i·11-s + (0.809 − 1.40i)13-s + (−0.535 − 0.309i)14-s − 16-s + (−1.40 − 0.809i)17-s + (−0.309 − 0.535i)19-s − 1.61·22-s + (−0.535 + 0.309i)23-s + (0.809 − 1.40i)25-s + (1.40 + 0.809i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.393291887\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.393291887\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 - iT - T^{2} \) |
| 5 | \( 1 + (-1.40 + 0.809i)T + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 - 1.61iT - T^{2} \) |
| 13 | \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (1.40 + 0.809i)T + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.535 - 0.309i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - 0.618iT - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (-0.535 + 0.309i)T + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + 0.618iT - T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - 1.61T + T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.935436817593949735026410755179, −9.135074913497067855148915191731, −8.604891829641042658644181894917, −7.56504646294156340155831746979, −6.67340880844829369424585729968, −5.99709904102110869506606644171, −5.27394560238537368144732068479, −4.59739993480483687202029735520, −2.64725495857878167579861452491, −1.85408965094289176994301350777,
1.54107757652419307287380007244, 2.36441836502158253006420749801, 3.42976903320511324576292803656, 4.19540062594556646775632826386, 6.04735708983201414059442426057, 6.26970353906060181851784600810, 7.01627413301465205729240159166, 8.566613953841310767559214957257, 9.197868698032742772079902862308, 10.15004872775300703780860495784