L(s) = 1 | − 3-s − 4·5-s − 3·7-s − 2·9-s + 5·11-s + 4·15-s − 6·17-s + 2·19-s + 3·21-s − 6·23-s + 11·25-s + 5·27-s − 6·29-s + 4·31-s − 5·33-s + 12·35-s + 37-s − 9·41-s + 4·43-s + 8·45-s − 7·47-s + 2·49-s + 6·51-s + 9·53-s − 20·55-s − 2·57-s − 4·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.78·5-s − 1.13·7-s − 2/3·9-s + 1.50·11-s + 1.03·15-s − 1.45·17-s + 0.458·19-s + 0.654·21-s − 1.25·23-s + 11/5·25-s + 0.962·27-s − 1.11·29-s + 0.718·31-s − 0.870·33-s + 2.02·35-s + 0.164·37-s − 1.40·41-s + 0.609·43-s + 1.19·45-s − 1.02·47-s + 2/7·49-s + 0.840·51-s + 1.23·53-s − 2.69·55-s − 0.264·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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
−12.04878843613669660208113250042, −11.80292650710754620844959955705, −10.84165866699062540441823894097, −9.340745919593799433087994357283, −8.373681526994173747684275212158, −7.02842485181405703141679387745, −6.19561582477303016221221714033, −4.37419194151079963211712663708, −3.39247583452853063224757971641, 0,
3.39247583452853063224757971641, 4.37419194151079963211712663708, 6.19561582477303016221221714033, 7.02842485181405703141679387745, 8.373681526994173747684275212158, 9.340745919593799433087994357283, 10.84165866699062540441823894097, 11.80292650710754620844959955705, 12.04878843613669660208113250042