| L(s) = 1 | + 5-s + 4·7-s − 11-s − 13-s − 8·17-s + 2·19-s + 25-s − 6·29-s + 8·31-s + 4·35-s − 8·37-s − 10·41-s − 4·43-s + 9·49-s + 6·53-s − 55-s − 4·59-s − 2·61-s − 65-s − 2·67-s + 12·71-s − 2·73-s − 4·77-s + 16·79-s + 4·83-s − 8·85-s + 18·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 0.301·11-s − 0.277·13-s − 1.94·17-s + 0.458·19-s + 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.676·35-s − 1.31·37-s − 1.56·41-s − 0.609·43-s + 9/7·49-s + 0.824·53-s − 0.134·55-s − 0.520·59-s − 0.256·61-s − 0.124·65-s − 0.244·67-s + 1.42·71-s − 0.234·73-s − 0.455·77-s + 1.80·79-s + 0.439·83-s − 0.867·85-s + 1.90·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.481572732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.481572732\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.65962070372035, −13.51519473623737, −12.80273144796034, −12.12275102699551, −11.75830962519073, −11.25081506680729, −10.84237182148852, −10.36196433127364, −9.856875634417893, −9.127694393800881, −8.777422559674917, −8.262342012864536, −7.814692870261616, −7.177257181813471, −6.685189960032559, −6.181033564770248, −5.348701839672733, −4.965355875340784, −4.681197476236644, −3.919846363402141, −3.251485244721591, −2.325955103037038, −2.031337704669162, −1.438456983877782, −0.4753217452297724,
0.4753217452297724, 1.438456983877782, 2.031337704669162, 2.325955103037038, 3.251485244721591, 3.919846363402141, 4.681197476236644, 4.965355875340784, 5.348701839672733, 6.181033564770248, 6.685189960032559, 7.177257181813471, 7.814692870261616, 8.262342012864536, 8.777422559674917, 9.127694393800881, 9.856875634417893, 10.36196433127364, 10.84237182148852, 11.25081506680729, 11.75830962519073, 12.12275102699551, 12.80273144796034, 13.51519473623737, 13.65962070372035
