| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s + 2·11-s − 15-s − 4·21-s + 8·23-s + 25-s − 27-s + 2·29-s + 2·31-s − 2·33-s + 4·35-s − 8·37-s − 2·41-s − 4·43-s + 45-s + 9·49-s − 6·53-s + 2·55-s + 14·59-s − 14·61-s + 4·63-s + 4·67-s − 8·69-s − 8·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s + 0.603·11-s − 0.258·15-s − 0.872·21-s + 1.66·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.359·31-s − 0.348·33-s + 0.676·35-s − 1.31·37-s − 0.312·41-s − 0.609·43-s + 0.149·45-s + 9/7·49-s − 0.824·53-s + 0.269·55-s + 1.82·59-s − 1.79·61-s + 0.503·63-s + 0.488·67-s − 0.963·69-s − 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.043375533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.043375533\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| 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
−9.051991937421236244849524256296, −8.552637465870575656105717069494, −7.54495099732933272728899482262, −6.83796031860810061922805918413, −5.96040992871476533750493888142, −4.99521209921548994066925274223, −4.66240850488548563222771309084, −3.34288207185341943924545769949, −1.95637318789282511149172740920, −1.09595735816982629557946257255,
1.09595735816982629557946257255, 1.95637318789282511149172740920, 3.34288207185341943924545769949, 4.66240850488548563222771309084, 4.99521209921548994066925274223, 5.96040992871476533750493888142, 6.83796031860810061922805918413, 7.54495099732933272728899482262, 8.552637465870575656105717069494, 9.051991937421236244849524256296
