L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s − 2·13-s + 15-s + 2·17-s + 19-s − 3·21-s − 2·23-s + 25-s − 27-s − 8·29-s + 5·31-s − 3·35-s − 11·37-s + 2·39-s + 6·41-s − 8·43-s − 45-s + 8·47-s + 2·49-s − 2·51-s + 2·53-s − 57-s + 4·59-s − 3·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 0.485·17-s + 0.229·19-s − 0.654·21-s − 0.417·23-s + 1/5·25-s − 0.192·27-s − 1.48·29-s + 0.898·31-s − 0.507·35-s − 1.80·37-s + 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s + 1.16·47-s + 2/7·49-s − 0.280·51-s + 0.274·53-s − 0.132·57-s + 0.520·59-s − 0.384·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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
−7.44250359041455009828178730643, −7.12208650206504981237159459538, −6.02001014423872765072063219804, −5.41225154460705369717017755426, −4.76922045058358849931203431395, −4.12365961823212500664594320788, −3.23148232607795206325839181385, −2.08983579544218622903909197994, −1.26111679792028208571336116550, 0,
1.26111679792028208571336116550, 2.08983579544218622903909197994, 3.23148232607795206325839181385, 4.12365961823212500664594320788, 4.76922045058358849931203431395, 5.41225154460705369717017755426, 6.02001014423872765072063219804, 7.12208650206504981237159459538, 7.44250359041455009828178730643