L(s) = 1 | − 3-s + 5-s − 2·7-s + 9-s + 13-s − 15-s − 2·19-s + 2·21-s − 2·23-s + 25-s − 27-s − 4·29-s + 4·31-s − 2·35-s − 2·37-s − 39-s + 6·41-s + 4·43-s + 45-s + 8·47-s − 3·49-s − 2·53-s + 2·57-s + 4·59-s + 2·61-s − 2·63-s + 65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.277·13-s − 0.258·15-s − 0.458·19-s + 0.436·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 0.742·29-s + 0.718·31-s − 0.338·35-s − 0.328·37-s − 0.160·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.264·57-s + 0.520·59-s + 0.256·61-s − 0.251·63-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 188760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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
−13.20831832999784, −12.85613288657974, −12.48488062667110, −12.00962568960763, −11.44466036106106, −10.88448387177516, −10.67475586404272, −9.913428951512745, −9.733342080938241, −9.185845517219368, −8.656937829695971, −8.092577702588699, −7.548598463579245, −6.922415170983143, −6.545701319486109, −6.080623151046030, −5.588840822768440, −5.212917085605760, −4.379069473319618, −4.018840150159494, −3.419230931315414, −2.663072497168513, −2.221424987658793, −1.429309719853279, −0.7578434945885616, 0,
0.7578434945885616, 1.429309719853279, 2.221424987658793, 2.663072497168513, 3.419230931315414, 4.018840150159494, 4.379069473319618, 5.212917085605760, 5.588840822768440, 6.080623151046030, 6.545701319486109, 6.922415170983143, 7.548598463579245, 8.092577702588699, 8.656937829695971, 9.185845517219368, 9.733342080938241, 9.913428951512745, 10.67475586404272, 10.88448387177516, 11.44466036106106, 12.00962568960763, 12.48488062667110, 12.85613288657974, 13.20831832999784