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 − 7·49-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 + 14·101-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 − 49-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 + 1.39·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.013504913\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.013504913\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 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
−9.547438999502026658807482308365, −8.667981626902712297406058340761, −8.176989849593365125562259771659, −6.68758445534728746261165546175, −6.54644616692428564421950720399, −5.41791684827641272932549263816, −4.40013024639620411659480510891, −3.53852999845818475814528031258, −2.28371327004468584640295662946, −1.09896676042479434195845174014,
1.09896676042479434195845174014, 2.28371327004468584640295662946, 3.53852999845818475814528031258, 4.40013024639620411659480510891, 5.41791684827641272932549263816, 6.54644616692428564421950720399, 6.68758445534728746261165546175, 8.176989849593365125562259771659, 8.667981626902712297406058340761, 9.547438999502026658807482308365