| L(s) = 1 | − 2·3-s − 3·5-s − 7-s + 9-s − 6·11-s + 2·13-s + 6·15-s + 6·17-s − 19-s + 2·21-s − 6·23-s + 4·25-s + 4·27-s + 31-s + 12·33-s + 3·35-s − 10·37-s − 4·39-s − 9·41-s + 8·43-s − 3·45-s − 6·49-s − 12·51-s + 18·55-s + 2·57-s − 3·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.554·13-s + 1.54·15-s + 1.45·17-s − 0.229·19-s + 0.436·21-s − 1.25·23-s + 4/5·25-s + 0.769·27-s + 0.179·31-s + 2.08·33-s + 0.507·35-s − 1.64·37-s − 0.640·39-s − 1.40·41-s + 1.21·43-s − 0.447·45-s − 6/7·49-s − 1.68·51-s + 2.42·55-s + 0.264·57-s − 0.390·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
| 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
−12.46026360788664430586581389482, −11.98385590607492315332591193960, −10.87563904560238509026342087447, −10.20308922870445519795206495827, −8.307924355052226010131976469241, −7.50902523830307538426781271510, −6.02777066127053067002590555832, −4.97533809273740302436990144721, −3.39823135309375851576061845233, 0,
3.39823135309375851576061845233, 4.97533809273740302436990144721, 6.02777066127053067002590555832, 7.50902523830307538426781271510, 8.307924355052226010131976469241, 10.20308922870445519795206495827, 10.87563904560238509026342087447, 11.98385590607492315332591193960, 12.46026360788664430586581389482
