| L(s) = 1 | − 2-s − 3-s − 4-s − 2·5-s + 6-s + 3·8-s + 9-s + 2·10-s + 12-s − 6·13-s + 2·15-s − 16-s − 6·17-s − 18-s + 2·20-s + 4·23-s − 3·24-s − 25-s + 6·26-s − 27-s − 2·29-s − 2·30-s − 8·31-s − 5·32-s + 6·34-s − 36-s + 10·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 1.66·13-s + 0.516·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.447·20-s + 0.834·23-s − 0.612·24-s − 1/5·25-s + 1.17·26-s − 0.192·27-s − 0.371·29-s − 0.365·30-s − 1.43·31-s − 0.883·32-s + 1.02·34-s − 1/6·36-s + 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1083 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3450518079\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3450518079\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.676043766169960369886673569255, −9.211204361001959275319801150204, −8.204224733564106943471891272665, −7.43259308735181576062150293394, −6.88281746236137995334549719629, −5.42974656732833448500123753190, −4.62864849822679992384179631378, −3.91841567788800315953565193679, −2.25179019930198933577153857080, −0.49711700208122723907813480784,
0.49711700208122723907813480784, 2.25179019930198933577153857080, 3.91841567788800315953565193679, 4.62864849822679992384179631378, 5.42974656732833448500123753190, 6.88281746236137995334549719629, 7.43259308735181576062150293394, 8.204224733564106943471891272665, 9.211204361001959275319801150204, 9.676043766169960369886673569255
