| L(s) = 1 | − 2-s + 2.24·3-s + 4-s − 5-s − 2.24·6-s + 0.778·7-s − 8-s + 2.05·9-s + 10-s + 5.19·11-s + 2.24·12-s − 3.83·13-s − 0.778·14-s − 2.24·15-s + 16-s + 5.30·17-s − 2.05·18-s + 3.71·19-s − 20-s + 1.75·21-s − 5.19·22-s + 7.30·23-s − 2.24·24-s + 25-s + 3.83·26-s − 2.11·27-s + 0.778·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.29·3-s + 0.5·4-s − 0.447·5-s − 0.918·6-s + 0.294·7-s − 0.353·8-s + 0.686·9-s + 0.316·10-s + 1.56·11-s + 0.649·12-s − 1.06·13-s − 0.208·14-s − 0.580·15-s + 0.250·16-s + 1.28·17-s − 0.485·18-s + 0.853·19-s − 0.223·20-s + 0.382·21-s − 1.10·22-s + 1.52·23-s − 0.459·24-s + 0.200·25-s + 0.752·26-s − 0.407·27-s + 0.147·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.551205292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.551205292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| good | 3 | \( 1 - 2.24T + 3T^{2} \) |
| 7 | \( 1 - 0.778T + 7T^{2} \) |
| 11 | \( 1 - 5.19T + 11T^{2} \) |
| 13 | \( 1 + 3.83T + 13T^{2} \) |
| 17 | \( 1 - 5.30T + 17T^{2} \) |
| 19 | \( 1 - 3.71T + 19T^{2} \) |
| 23 | \( 1 - 7.30T + 23T^{2} \) |
| 29 | \( 1 + 9.52T + 29T^{2} \) |
| 31 | \( 1 + 0.412T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 8.66T + 41T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 53 | \( 1 + 4.05T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 + 13.9T + 67T^{2} \) |
| 71 | \( 1 - 1.19T + 71T^{2} \) |
| 73 | \( 1 + 9.85T + 73T^{2} \) |
| 79 | \( 1 - 4.08T + 79T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 + 10.8T + 97T^{2} \) |
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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
−11.17894856711646831125497368368, −9.820665683799804300756520580307, −9.271358355941244493737534813177, −8.570252140706039296530331760444, −7.51272775311896320618846045716, −7.11686656238056242629556476569, −5.42704304791645152829253897350, −3.86231430787676711132163894953, −2.96390400231112389804841259598, −1.47661923668323543711792828119,
1.47661923668323543711792828119, 2.96390400231112389804841259598, 3.86231430787676711132163894953, 5.42704304791645152829253897350, 7.11686656238056242629556476569, 7.51272775311896320618846045716, 8.570252140706039296530331760444, 9.271358355941244493737534813177, 9.820665683799804300756520580307, 11.17894856711646831125497368368
