| L(s) = 1 | + 2-s − 3-s + 4-s − 3.20·5-s − 6-s − 2.02·7-s + 8-s + 9-s − 3.20·10-s − 1.72·11-s − 12-s + 13-s − 2.02·14-s + 3.20·15-s + 16-s − 0.669·17-s + 18-s + 6.70·19-s − 3.20·20-s + 2.02·21-s − 1.72·22-s + 2.38·23-s − 24-s + 5.30·25-s + 26-s − 27-s − 2.02·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.43·5-s − 0.408·6-s − 0.764·7-s + 0.353·8-s + 0.333·9-s − 1.01·10-s − 0.520·11-s − 0.288·12-s + 0.277·13-s − 0.540·14-s + 0.828·15-s + 0.250·16-s − 0.162·17-s + 0.235·18-s + 1.53·19-s − 0.717·20-s + 0.441·21-s − 0.367·22-s + 0.497·23-s − 0.204·24-s + 1.06·25-s + 0.196·26-s − 0.192·27-s − 0.382·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 3.20T + 5T^{2} \) |
| 7 | \( 1 + 2.02T + 7T^{2} \) |
| 11 | \( 1 + 1.72T + 11T^{2} \) |
| 17 | \( 1 + 0.669T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 + 2.29T + 29T^{2} \) |
| 31 | \( 1 + 4.65T + 31T^{2} \) |
| 37 | \( 1 + 2.89T + 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 - 3.74T + 43T^{2} \) |
| 47 | \( 1 - 8.51T + 47T^{2} \) |
| 53 | \( 1 - 11.6T + 53T^{2} \) |
| 59 | \( 1 + 3.76T + 59T^{2} \) |
| 61 | \( 1 + 3.46T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 6.69T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 0.941T + 79T^{2} \) |
| 83 | \( 1 - 0.0679T + 83T^{2} \) |
| 89 | \( 1 - 15.8T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.42763184138262740473399591314, −6.85086682735769345369101427422, −5.99553931374322730112049238437, −5.32537018074000692504635584530, −4.70520293657170674964051555548, −3.73716823341925807115252057899, −3.47695965556856531049337197028, −2.51258197097722731059887072870, −1.07306544798497275066455203342, 0,
1.07306544798497275066455203342, 2.51258197097722731059887072870, 3.47695965556856531049337197028, 3.73716823341925807115252057899, 4.70520293657170674964051555548, 5.32537018074000692504635584530, 5.99553931374322730112049238437, 6.85086682735769345369101427422, 7.42763184138262740473399591314
