L(s) = 1 | + 2.26i·3-s − 1.11·7-s − 2.11·9-s + 5.37i·11-s + (1.26 + 3.37i)13-s − 7.90i·19-s − 2.52i·21-s + 6.49i·23-s + 2i·27-s − 3.63·29-s − 3.14i·31-s − 12.1·33-s − 7.40·37-s + (−7.63 + 2.85i)39-s − 4.75i·41-s + ⋯ |
L(s) = 1 | + 1.30i·3-s − 0.421·7-s − 0.705·9-s + 1.62i·11-s + (0.349 + 0.936i)13-s − 1.81i·19-s − 0.550i·21-s + 1.35i·23-s + 0.384i·27-s − 0.675·29-s − 0.565i·31-s − 2.11·33-s − 1.21·37-s + (−1.22 + 0.456i)39-s − 0.742i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.144170922\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.144170922\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-1.26 - 3.37i)T \) |
good | 3 | \( 1 - 2.26iT - 3T^{2} \) |
| 7 | \( 1 + 1.11T + 7T^{2} \) |
| 11 | \( 1 - 5.37iT - 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 7.90iT - 19T^{2} \) |
| 23 | \( 1 - 6.49iT - 23T^{2} \) |
| 29 | \( 1 + 3.63T + 29T^{2} \) |
| 31 | \( 1 + 3.14iT - 31T^{2} \) |
| 37 | \( 1 + 7.40T + 37T^{2} \) |
| 41 | \( 1 + 4.75iT - 41T^{2} \) |
| 43 | \( 1 - 4.78iT - 43T^{2} \) |
| 47 | \( 1 - 6.16T + 47T^{2} \) |
| 53 | \( 1 + 0.292iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 5.34T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 3.43iT - 71T^{2} \) |
| 73 | \( 1 - 4.59T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 + 7.40T + 83T^{2} \) |
| 89 | \( 1 + 14.8iT - 89T^{2} \) |
| 97 | \( 1 - 3.76T + 97T^{2} \) |
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
−9.855558250328310626631954536717, −9.341922826720273073623374225392, −8.902353214182896062086944893182, −7.38953922051431564580721819906, −6.92569679147410576981838048034, −5.67383958333756034565978141062, −4.70943761193993475291737210290, −4.21844171348275325713469279904, −3.19889958733815877689541497328, −1.88563042287382903623469608333,
0.47209481027513463578094777977, 1.63202583425899656706702994114, 2.96222359943498307246660196235, 3.75020927033707863401052893757, 5.38542608859915520552710335800, 6.12828531950334094057949542178, 6.63846734079698453589132482879, 7.82246148019469434120786964036, 8.191676132330399078956640761171, 8.999662598957129433292253284077