| L(s) = 1 | − 3·5-s − 7-s + 6·11-s − 4·13-s − 3·17-s − 2·19-s − 6·23-s + 4·25-s + 6·29-s + 4·31-s + 3·35-s − 7·37-s + 3·41-s + 43-s + 9·47-s + 49-s + 6·53-s − 18·55-s + 9·59-s − 10·61-s + 12·65-s + 4·67-s + 2·73-s − 6·77-s + 79-s + 3·83-s + 9·85-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 0.377·7-s + 1.80·11-s − 1.10·13-s − 0.727·17-s − 0.458·19-s − 1.25·23-s + 4/5·25-s + 1.11·29-s + 0.718·31-s + 0.507·35-s − 1.15·37-s + 0.468·41-s + 0.152·43-s + 1.31·47-s + 1/7·49-s + 0.824·53-s − 2.42·55-s + 1.17·59-s − 1.28·61-s + 1.48·65-s + 0.488·67-s + 0.234·73-s − 0.683·77-s + 0.112·79-s + 0.329·83-s + 0.976·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.078499554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.078499554\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−8.644890615125473184723296713633, −8.048911376131231568860835984621, −7.04562425666546931796340536619, −6.74621439114006943593472598781, −5.76520801586940676795776197507, −4.35727880808405451130327441222, −4.24558244070364871306815689163, −3.25082978104382749561512942653, −2.10371002829409123939002347931, −0.61846901235906980408646092164,
0.61846901235906980408646092164, 2.10371002829409123939002347931, 3.25082978104382749561512942653, 4.24558244070364871306815689163, 4.35727880808405451130327441222, 5.76520801586940676795776197507, 6.74621439114006943593472598781, 7.04562425666546931796340536619, 8.048911376131231568860835984621, 8.644890615125473184723296713633
