L(s) = 1 | − 5-s + 4·11-s − 2·13-s − 2·17-s + 8·19-s + 4·23-s + 25-s + 6·29-s + 2·37-s − 6·41-s − 4·43-s + 12·47-s + 6·53-s − 4·55-s − 12·59-s − 14·61-s + 2·65-s + 12·67-s − 2·73-s + 8·79-s + 4·83-s + 2·85-s + 2·89-s − 8·95-s + 14·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.20·11-s − 0.554·13-s − 0.485·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.328·37-s − 0.937·41-s − 0.609·43-s + 1.75·47-s + 0.824·53-s − 0.539·55-s − 1.56·59-s − 1.79·61-s + 0.248·65-s + 1.46·67-s − 0.234·73-s + 0.900·79-s + 0.439·83-s + 0.216·85-s + 0.211·89-s − 0.820·95-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.775490054\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.775490054\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 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 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.05310622753734, −13.71120362052070, −13.27502952125721, −12.37251573942089, −12.01333723501593, −11.88500044921548, −11.09449775298599, −10.74358446455230, −10.01448874911835, −9.469941441279118, −9.130282499914449, −8.566577756136220, −7.950768424237361, −7.297414191115221, −7.020259697363216, −6.399255504187290, −5.791911647208640, −5.026579779264232, −4.669036921252898, −3.993288531899463, −3.303018727412050, −2.913509826369104, −2.003882153631970, −1.178786763818570, −0.6294512850978534,
0.6294512850978534, 1.178786763818570, 2.003882153631970, 2.913509826369104, 3.303018727412050, 3.993288531899463, 4.669036921252898, 5.026579779264232, 5.791911647208640, 6.399255504187290, 7.020259697363216, 7.297414191115221, 7.950768424237361, 8.566577756136220, 9.130282499914449, 9.469941441279118, 10.01448874911835, 10.74358446455230, 11.09449775298599, 11.88500044921548, 12.01333723501593, 12.37251573942089, 13.27502952125721, 13.71120362052070, 14.05310622753734