L(s) = 1 | + (0.366 + 1.36i)2-s − 1.73·3-s + (−1.73 + i)4-s + i·5-s + (−0.633 − 2.36i)6-s + (−1.73 + 2i)7-s + (−2 − 1.99i)8-s + (−1.36 + 0.366i)10-s + 0.267i·11-s + (2.99 − 1.73i)12-s + 0.464i·13-s + (−3.36 − 1.63i)14-s − 1.73i·15-s + (1.99 − 3.46i)16-s + 6.46i·17-s + ⋯ |
L(s) = 1 | + (0.258 + 0.965i)2-s − 1.00·3-s + (−0.866 + 0.5i)4-s + 0.447i·5-s + (−0.258 − 0.965i)6-s + (−0.654 + 0.755i)7-s + (−0.707 − 0.707i)8-s + (−0.431 + 0.115i)10-s + 0.0807i·11-s + (0.866 − 0.499i)12-s + 0.128i·13-s + (−0.899 − 0.436i)14-s − 0.447i·15-s + (0.499 − 0.866i)16-s + 1.56i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0589062 + 0.617788i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0589062 + 0.617788i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.366 - 1.36i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
good | 3 | \( 1 + 1.73T + 3T^{2} \) |
| 11 | \( 1 - 0.267iT - 11T^{2} \) |
| 13 | \( 1 - 0.464iT - 13T^{2} \) |
| 17 | \( 1 - 6.46iT - 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 - 1.46iT - 23T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 9.46T + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 1.73T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 9.46iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 7.46iT - 71T^{2} \) |
| 73 | \( 1 + 12.9iT - 73T^{2} \) |
| 79 | \( 1 - 14.6iT - 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 + 2.53iT - 89T^{2} \) |
| 97 | \( 1 - 13.3iT - 97T^{2} \) |
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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
−13.72871571461879644589810260012, −12.52931094571205585742381997523, −11.91930144487131302115575156422, −10.58583817224558325641605871202, −9.404360551564499699705224351273, −8.228199031050160078963874641245, −6.81842044350380925908750047663, −6.02985952146808533006216944611, −5.16220767501999316703122289698, −3.39919770862336868549693868751,
0.66756306154638102591564443936, 3.17950822102262844129885461979, 4.75297579295402906626388701752, 5.65956700888107471052541921644, 7.11378647964160262014460832949, 8.868877961939504159867187409771, 9.917001754986770221064083038267, 10.77788176836264166678661290052, 11.79678084710694179259016051729, 12.35286703727573479094863423883