L(s) = 1 | + 0.369·3-s − 3.94·5-s + 3.05·7-s − 2.86·9-s + 5.92·11-s + 4.39·13-s − 1.45·15-s + 6.07·17-s + 1.60·19-s + 1.12·21-s + 7.25·23-s + 10.5·25-s − 2.16·27-s + 2.83·29-s − 6.95·31-s + 2.18·33-s − 12.0·35-s + 7.42·37-s + 1.62·39-s − 3.32·41-s + 1.53·43-s + 11.2·45-s − 3.21·47-s + 2.34·49-s + 2.24·51-s − 9.95·53-s − 23.3·55-s + ⋯ |
L(s) = 1 | + 0.213·3-s − 1.76·5-s + 1.15·7-s − 0.954·9-s + 1.78·11-s + 1.21·13-s − 0.376·15-s + 1.47·17-s + 0.368·19-s + 0.246·21-s + 1.51·23-s + 2.11·25-s − 0.416·27-s + 0.525·29-s − 1.24·31-s + 0.381·33-s − 2.03·35-s + 1.22·37-s + 0.260·39-s − 0.519·41-s + 0.233·43-s + 1.68·45-s − 0.469·47-s + 0.335·49-s + 0.314·51-s − 1.36·53-s − 3.15·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.338055552\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.338055552\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 3 | \( 1 - 0.369T + 3T^{2} \) |
| 5 | \( 1 + 3.94T + 5T^{2} \) |
| 7 | \( 1 - 3.05T + 7T^{2} \) |
| 11 | \( 1 - 5.92T + 11T^{2} \) |
| 13 | \( 1 - 4.39T + 13T^{2} \) |
| 17 | \( 1 - 6.07T + 17T^{2} \) |
| 19 | \( 1 - 1.60T + 19T^{2} \) |
| 23 | \( 1 - 7.25T + 23T^{2} \) |
| 29 | \( 1 - 2.83T + 29T^{2} \) |
| 31 | \( 1 + 6.95T + 31T^{2} \) |
| 37 | \( 1 - 7.42T + 37T^{2} \) |
| 41 | \( 1 + 3.32T + 41T^{2} \) |
| 43 | \( 1 - 1.53T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 + 9.95T + 53T^{2} \) |
| 59 | \( 1 - 3.98T + 59T^{2} \) |
| 61 | \( 1 - 9.96T + 61T^{2} \) |
| 67 | \( 1 + 7.23T + 67T^{2} \) |
| 71 | \( 1 - 2.54T + 71T^{2} \) |
| 73 | \( 1 - 9.26T + 73T^{2} \) |
| 79 | \( 1 + 16.8T + 79T^{2} \) |
| 83 | \( 1 - 18.0T + 83T^{2} \) |
| 89 | \( 1 + 8.06T + 89T^{2} \) |
| 97 | \( 1 - 0.446T + 97T^{2} \) |
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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
−8.103327806969307558723052128091, −7.26223588067615973080533906470, −6.61573663653969096813845028826, −5.67536744241823674744483171938, −4.93060993429674939085976949145, −4.10825252450404377280849691842, −3.55154072647261150721123561439, −3.04472898246792798159785809344, −1.44060323797923623622684643978, −0.864117273812423011919062680864,
0.864117273812423011919062680864, 1.44060323797923623622684643978, 3.04472898246792798159785809344, 3.55154072647261150721123561439, 4.10825252450404377280849691842, 4.93060993429674939085976949145, 5.67536744241823674744483171938, 6.61573663653969096813845028826, 7.26223588067615973080533906470, 8.103327806969307558723052128091