| L(s) = 1 | + 2.00·3-s + 5-s + 1.43·7-s + 1.00·9-s − 5.83·11-s + 5.47·13-s + 2.00·15-s + 5.07·17-s − 4.93·19-s + 2.86·21-s + 5.36·23-s + 25-s − 3.99·27-s + 1.02·29-s + 6.94·31-s − 11.6·33-s + 1.43·35-s + 2.38·37-s + 10.9·39-s + 11.2·41-s + 3.38·43-s + 1.00·45-s − 7.43·47-s − 4.95·49-s + 10.1·51-s + 4.16·53-s − 5.83·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.540·7-s + 0.333·9-s − 1.75·11-s + 1.51·13-s + 0.516·15-s + 1.23·17-s − 1.13·19-s + 0.624·21-s + 1.11·23-s + 0.200·25-s − 0.769·27-s + 0.189·29-s + 1.24·31-s − 2.03·33-s + 0.241·35-s + 0.391·37-s + 1.75·39-s + 1.75·41-s + 0.516·43-s + 0.149·45-s − 1.08·47-s − 0.707·49-s + 1.42·51-s + 0.571·53-s − 0.786·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.776758993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.776758993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 2.00T + 3T^{2} \) |
| 7 | \( 1 - 1.43T + 7T^{2} \) |
| 11 | \( 1 + 5.83T + 11T^{2} \) |
| 13 | \( 1 - 5.47T + 13T^{2} \) |
| 17 | \( 1 - 5.07T + 17T^{2} \) |
| 19 | \( 1 + 4.93T + 19T^{2} \) |
| 23 | \( 1 - 5.36T + 23T^{2} \) |
| 29 | \( 1 - 1.02T + 29T^{2} \) |
| 31 | \( 1 - 6.94T + 31T^{2} \) |
| 37 | \( 1 - 2.38T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 3.38T + 43T^{2} \) |
| 47 | \( 1 + 7.43T + 47T^{2} \) |
| 53 | \( 1 - 4.16T + 53T^{2} \) |
| 59 | \( 1 - 0.918T + 59T^{2} \) |
| 61 | \( 1 + 7.71T + 61T^{2} \) |
| 67 | \( 1 - 8.10T + 67T^{2} \) |
| 71 | \( 1 + 6.03T + 71T^{2} \) |
| 73 | \( 1 - 3.67T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 8.84T + 83T^{2} \) |
| 89 | \( 1 - 7.39T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.84787807050582341478878363756, −7.55754364583255957128295023051, −6.33729925238176310092327392650, −5.79279734812908227581618549331, −5.01338401673991367981412085109, −4.23433981706594131318498588606, −3.21345301074236218196147903006, −2.77697139986393174179617700819, −1.95579022159549394575570785785, −0.933293087738491761185745954004,
0.933293087738491761185745954004, 1.95579022159549394575570785785, 2.77697139986393174179617700819, 3.21345301074236218196147903006, 4.23433981706594131318498588606, 5.01338401673991367981412085109, 5.79279734812908227581618549331, 6.33729925238176310092327392650, 7.55754364583255957128295023051, 7.84787807050582341478878363756
