| L(s) = 1 | − 2·7-s − 3·9-s − 11-s + 4·13-s + 4·17-s − 6·29-s + 2·37-s + 6·41-s + 2·43-s − 3·49-s + 10·53-s − 12·59-s − 6·61-s + 6·63-s − 12·67-s − 16·71-s − 4·73-s + 2·77-s + 4·79-s + 9·81-s + 2·83-s + 6·89-s − 8·91-s + 2·97-s + 3·99-s + 6·101-s + 4·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 9-s − 0.301·11-s + 1.10·13-s + 0.970·17-s − 1.11·29-s + 0.328·37-s + 0.937·41-s + 0.304·43-s − 3/7·49-s + 1.37·53-s − 1.56·59-s − 0.768·61-s + 0.755·63-s − 1.46·67-s − 1.89·71-s − 0.468·73-s + 0.227·77-s + 0.450·79-s + 81-s + 0.219·83-s + 0.635·89-s − 0.838·91-s + 0.203·97-s + 0.301·99-s + 0.597·101-s + 0.394·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| 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
−7.898936570941772463430241678285, −7.44620724213555121019017614969, −6.23862547700277970583253282336, −5.97420845383247042083925636614, −5.19222357781999791105637954091, −4.04649274617805365151136230133, −3.30588833879895129178850964400, −2.64602945319110350841739354126, −1.31982113773072061232939185969, 0,
1.31982113773072061232939185969, 2.64602945319110350841739354126, 3.30588833879895129178850964400, 4.04649274617805365151136230133, 5.19222357781999791105637954091, 5.97420845383247042083925636614, 6.23862547700277970583253282336, 7.44620724213555121019017614969, 7.898936570941772463430241678285
