L(s) = 1 | + 2·3-s − 5-s + 7-s + 9-s − 3·11-s − 4·13-s − 2·15-s + 3·17-s − 2·19-s + 2·21-s − 6·23-s + 25-s − 4·27-s + 3·29-s − 5·31-s − 6·33-s − 35-s + 37-s − 8·39-s + 3·41-s + 43-s − 45-s − 12·47-s − 6·49-s + 6·51-s + 3·53-s + 3·55-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s − 1.10·13-s − 0.516·15-s + 0.727·17-s − 0.458·19-s + 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.769·27-s + 0.557·29-s − 0.898·31-s − 1.04·33-s − 0.169·35-s + 0.164·37-s − 1.28·39-s + 0.468·41-s + 0.152·43-s − 0.149·45-s − 1.75·47-s − 6/7·49-s + 0.840·51-s + 0.412·53-s + 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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
−8.213079647180113859147339692981, −7.80191348675256968137318821157, −7.25190206823653094417112473387, −6.05862458218504358555244114858, −5.14462937098489056863036116363, −4.35398309815315781351355776419, −3.40211614659968515387465276614, −2.65209192124627030493117144784, −1.82892205192733241717251603919, 0,
1.82892205192733241717251603919, 2.65209192124627030493117144784, 3.40211614659968515387465276614, 4.35398309815315781351355776419, 5.14462937098489056863036116363, 6.05862458218504358555244114858, 7.25190206823653094417112473387, 7.80191348675256968137318821157, 8.213079647180113859147339692981