| L(s) = 1 | − 2.56·3-s − 7-s + 3.56·9-s + 2.12·11-s − 2·13-s + 2.56·17-s − 0.561·19-s + 2.56·21-s − 5.56·23-s − 1.43·27-s + 7.56·29-s − 0.876·31-s − 5.43·33-s + 11.8·37-s + 5.12·39-s − 6.56·41-s − 2.43·43-s − 8.24·47-s + 49-s − 6.56·51-s − 7.12·53-s + 1.43·57-s − 13.3·59-s + 2.87·61-s − 3.56·63-s − 16.1·67-s + 14.2·69-s + ⋯ |
| L(s) = 1 | − 1.47·3-s − 0.377·7-s + 1.18·9-s + 0.640·11-s − 0.554·13-s + 0.621·17-s − 0.128·19-s + 0.558·21-s − 1.15·23-s − 0.276·27-s + 1.40·29-s − 0.157·31-s − 0.946·33-s + 1.94·37-s + 0.820·39-s − 1.02·41-s − 0.371·43-s − 1.20·47-s + 0.142·49-s − 0.918·51-s − 0.978·53-s + 0.190·57-s − 1.74·59-s + 0.368·61-s − 0.448·63-s − 1.96·67-s + 1.71·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 + 0.561T + 19T^{2} \) |
| 23 | \( 1 + 5.56T + 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 + 0.876T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 6.56T + 41T^{2} \) |
| 43 | \( 1 + 2.43T + 43T^{2} \) |
| 47 | \( 1 + 8.24T + 47T^{2} \) |
| 53 | \( 1 + 7.12T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 2.87T + 61T^{2} \) |
| 67 | \( 1 + 16.1T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 6.68T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.448236076997645326736019268313, −8.268059032052610293781751519605, −7.36629948391706331494942942697, −6.33825622347197078298359905109, −6.09437465928392248430617542934, −4.97851328967325605875195873002, −4.29528314099125389739771012556, −3.00158667861434916034148995806, −1.39586011083815699631381559088, 0,
1.39586011083815699631381559088, 3.00158667861434916034148995806, 4.29528314099125389739771012556, 4.97851328967325605875195873002, 6.09437465928392248430617542934, 6.33825622347197078298359905109, 7.36629948391706331494942942697, 8.268059032052610293781751519605, 9.448236076997645326736019268313
