| L(s) = 1 | − 2-s − 4-s + 3·8-s − 3·9-s − 11-s − 2·13-s − 16-s − 6·17-s + 3·18-s − 4·19-s + 22-s − 4·23-s + 2·26-s + 6·29-s − 8·31-s − 5·32-s + 6·34-s + 3·36-s + 2·37-s + 4·38-s + 2·41-s − 4·43-s + 44-s + 4·46-s + 12·47-s − 7·49-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 9-s − 0.301·11-s − 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.917·19-s + 0.213·22-s − 0.834·23-s + 0.392·26-s + 1.11·29-s − 1.43·31-s − 0.883·32-s + 1.02·34-s + 1/2·36-s + 0.328·37-s + 0.648·38-s + 0.312·41-s − 0.609·43-s + 0.150·44-s + 0.589·46-s + 1.75·47-s − 49-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−11.13962481139607451618080355242, −10.44930765910572775087302550356, −9.336297212975655357629362917616, −8.631003161371790257232477095324, −7.82885193609770565758085877155, −6.54918508631964327843106616026, −5.23353820210759730524176244525, −4.11816586584967359873614255585, −2.31804956482818418780108048277, 0,
2.31804956482818418780108048277, 4.11816586584967359873614255585, 5.23353820210759730524176244525, 6.54918508631964327843106616026, 7.82885193609770565758085877155, 8.631003161371790257232477095324, 9.336297212975655357629362917616, 10.44930765910572775087302550356, 11.13962481139607451618080355242
