| L(s) = 1 | + 5-s − 2·7-s + 4·11-s − 13-s − 8·17-s + 6·19-s + 6·23-s + 25-s + 4·29-s − 2·35-s − 2·37-s + 2·41-s + 4·43-s − 3·49-s + 10·53-s + 4·55-s + 4·59-s − 10·61-s − 65-s − 12·67-s − 8·71-s − 8·73-s − 8·77-s − 8·79-s + 12·83-s − 8·85-s + 14·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s + 1.20·11-s − 0.277·13-s − 1.94·17-s + 1.37·19-s + 1.25·23-s + 1/5·25-s + 0.742·29-s − 0.338·35-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 3/7·49-s + 1.37·53-s + 0.539·55-s + 0.520·59-s − 1.28·61-s − 0.124·65-s − 1.46·67-s − 0.949·71-s − 0.936·73-s − 0.911·77-s − 0.900·79-s + 1.31·83-s − 0.867·85-s + 1.48·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.132313219\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.132313219\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−7.44402085815186928221316548841, −6.97292458083943566285937572696, −6.40072740812642782744428493784, −5.80778030856188485702098155254, −4.84659241124141227396461053825, −4.29896194556588441618526638864, −3.32325696980069181275040043023, −2.71173310432285834822225949996, −1.70630519538430677611499253161, −0.71285530202580182453173395478,
0.71285530202580182453173395478, 1.70630519538430677611499253161, 2.71173310432285834822225949996, 3.32325696980069181275040043023, 4.29896194556588441618526638864, 4.84659241124141227396461053825, 5.80778030856188485702098155254, 6.40072740812642782744428493784, 6.97292458083943566285937572696, 7.44402085815186928221316548841
