L(s) = 1 | − 3-s + 1.41·7-s + 9-s + 1.41·13-s − 6·17-s + 6·19-s − 1.41·21-s − 8.48·23-s − 5·25-s − 27-s − 8.48·29-s − 1.41·31-s + 7.07·37-s − 1.41·39-s − 6·41-s + 6·43-s + 8.48·47-s − 5·49-s + 6·51-s + 8.48·53-s − 6·57-s − 7.07·61-s + 1.41·63-s − 4·67-s + 8.48·69-s + 8.48·71-s + 5·75-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.534·7-s + 0.333·9-s + 0.392·13-s − 1.45·17-s + 1.37·19-s − 0.308·21-s − 1.76·23-s − 25-s − 0.192·27-s − 1.57·29-s − 0.254·31-s + 1.16·37-s − 0.226·39-s − 0.937·41-s + 0.914·43-s + 1.23·47-s − 0.714·49-s + 0.840·51-s + 1.16·53-s − 0.794·57-s − 0.905·61-s + 0.178·63-s − 0.488·67-s + 1.02·69-s + 1.00·71-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
good | 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 7.07T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 8.48T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 1.41T + 79T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 12T + 97T^{2} \) |
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
−8.182569134092054322740174364679, −7.62101315787444188548272721275, −6.82131644085819383422332745295, −5.87214165405339079188319433737, −5.46263716681250846425108726051, −4.33157204055834791528361443964, −3.82755890513892064136854290855, −2.40205626345514359307191075707, −1.48414855806448574462850955510, 0,
1.48414855806448574462850955510, 2.40205626345514359307191075707, 3.82755890513892064136854290855, 4.33157204055834791528361443964, 5.46263716681250846425108726051, 5.87214165405339079188319433737, 6.82131644085819383422332745295, 7.62101315787444188548272721275, 8.182569134092054322740174364679