L(s) = 1 | − 3-s − 2.85·5-s − 7-s + 9-s − 3·11-s − 2.61·13-s + 2.85·15-s + 7.47·17-s + 4.23·19-s + 21-s + 23-s + 3.14·25-s − 27-s − 5·29-s − 10.2·31-s + 3·33-s + 2.85·35-s + 0.236·37-s + 2.61·39-s + 3·41-s − 0.618·43-s − 2.85·45-s − 1.23·47-s + 49-s − 7.47·51-s + 13.3·53-s + 8.56·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.27·5-s − 0.377·7-s + 0.333·9-s − 0.904·11-s − 0.726·13-s + 0.736·15-s + 1.81·17-s + 0.971·19-s + 0.218·21-s + 0.208·23-s + 0.629·25-s − 0.192·27-s − 0.928·29-s − 1.83·31-s + 0.522·33-s + 0.482·35-s + 0.0388·37-s + 0.419·39-s + 0.468·41-s − 0.0942·43-s − 0.425·45-s − 0.180·47-s + 0.142·49-s − 1.04·51-s + 1.83·53-s + 1.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 2.85T + 5T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 - 7.47T + 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 0.236T + 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 0.618T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 6.56T + 59T^{2} \) |
| 61 | \( 1 - 1.85T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 5.38T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 0.527T + 79T^{2} \) |
| 83 | \( 1 + 7.76T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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
−7.49008173233778470789614599497, −7.17868561786525938706492145498, −5.96297157922358057641276922706, −5.35511326913208161667877529131, −4.86982856125006096335192413129, −3.66364944026120324759645749448, −3.46614857300702303168685683394, −2.30513945316368675781818931338, −0.932518917413913557887006349079, 0,
0.932518917413913557887006349079, 2.30513945316368675781818931338, 3.46614857300702303168685683394, 3.66364944026120324759645749448, 4.86982856125006096335192413129, 5.35511326913208161667877529131, 5.96297157922358057641276922706, 7.17868561786525938706492145498, 7.49008173233778470789614599497