| L(s) = 1 | − 7-s − 3·11-s − 13-s − 5·17-s − 8·19-s + 2·23-s + 29-s − 2·31-s − 10·37-s + 6·41-s + 4·43-s + 11·47-s + 49-s + 6·53-s + 10·59-s + 10·67-s + 10·73-s + 3·77-s − 7·79-s + 12·83-s − 8·89-s + 91-s − 3·97-s + 12·101-s + 5·103-s + 8·107-s − 7·109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.904·11-s − 0.277·13-s − 1.21·17-s − 1.83·19-s + 0.417·23-s + 0.185·29-s − 0.359·31-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 1.60·47-s + 1/7·49-s + 0.824·53-s + 1.30·59-s + 1.22·67-s + 1.17·73-s + 0.341·77-s − 0.787·79-s + 1.31·83-s − 0.847·89-s + 0.104·91-s − 0.304·97-s + 1.19·101-s + 0.492·103-s + 0.773·107-s − 0.670·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.093994907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.093994907\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−8.134417703760000862320171648003, −7.17531866000457030845146971606, −6.75335303747783836428943890032, −5.91713151770727551400834388302, −5.19121269013267348545942151284, −4.38199065468322356767890565312, −3.71876572007764561674276863501, −2.53357081882484227580521751470, −2.13082614682211648213428780633, −0.51143467196140046597297738372,
0.51143467196140046597297738372, 2.13082614682211648213428780633, 2.53357081882484227580521751470, 3.71876572007764561674276863501, 4.38199065468322356767890565312, 5.19121269013267348545942151284, 5.91713151770727551400834388302, 6.75335303747783836428943890032, 7.17531866000457030845146971606, 8.134417703760000862320171648003
