L(s) = 1 | + (0.294 + 1.38i)2-s + (0.104 + 0.994i)3-s + (−0.913 + 0.406i)4-s + (0.978 + 0.207i)5-s + (−1.34 + 0.437i)6-s + (−1.40 − 0.147i)7-s + (−0.978 + 0.207i)9-s + 1.41i·10-s + (−0.499 − 0.866i)12-s + (−0.209 − 1.98i)14-s + (−0.104 + 0.994i)15-s + (−0.669 + 0.743i)16-s + (−0.575 − 1.29i)18-s + (−0.978 + 0.207i)20-s − 1.41i·21-s + ⋯ |
L(s) = 1 | + (0.294 + 1.38i)2-s + (0.104 + 0.994i)3-s + (−0.913 + 0.406i)4-s + (0.978 + 0.207i)5-s + (−1.34 + 0.437i)6-s + (−1.40 − 0.147i)7-s + (−0.978 + 0.207i)9-s + 1.41i·10-s + (−0.499 − 0.866i)12-s + (−0.209 − 1.98i)14-s + (−0.104 + 0.994i)15-s + (−0.669 + 0.743i)16-s + (−0.575 − 1.29i)18-s + (−0.978 + 0.207i)20-s − 1.41i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0804i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.215094054\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.215094054\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.294 - 1.38i)T + (-0.913 + 0.406i)T^{2} \) |
| 5 | \( 1 + (-0.978 - 0.207i)T + (0.913 + 0.406i)T^{2} \) |
| 7 | \( 1 + (1.40 + 0.147i)T + (0.978 + 0.207i)T^{2} \) |
| 13 | \( 1 + (0.104 - 0.994i)T^{2} \) |
| 17 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (-1.40 - 0.147i)T + (0.978 + 0.207i)T^{2} \) |
| 31 | \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (0.978 - 0.207i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \) |
| 53 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 61 | \( 1 + (-1.05 - 0.946i)T + (0.104 + 0.994i)T^{2} \) |
| 67 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.831 + 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 83 | \( 1 + (-1.05 - 0.946i)T + (0.104 + 0.994i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.978 - 0.207i)T + (0.913 - 0.406i)T^{2} \) |
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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
−10.06176808409886571986978124142, −9.746153939832171650875288206815, −8.792662668515614363534157571638, −8.035875755882814729459533903350, −6.64313405272068389527905242861, −6.43634546070287064711098911087, −5.51009992266980014430052475859, −4.73930342774084517571588061707, −3.59960799905316520659475671597, −2.53975633314205746964399173467,
1.03721026632452037312339256611, 2.33830857970292632373847837164, 2.84850785346325545104655531480, 3.98444549512726115950008617039, 5.39748251262291987750017796727, 6.29201416496053131364128384547, 6.89145858436435959260724100308, 8.139014446007905667041791877221, 9.255360993730004031164048611441, 9.714353812978840469588780155132