| L(s) = 1 | − 3.41·3-s − 5-s + 0.311·7-s + 8.63·9-s − 4.73·11-s − 4.87·13-s + 3.41·15-s + 5.19·17-s + 5.28·19-s − 1.06·21-s − 7.51·23-s + 25-s − 19.2·27-s + 1.85·29-s + 4.70·31-s + 16.1·33-s − 0.311·35-s − 3.55·37-s + 16.6·39-s + 9.83·41-s − 8.90·43-s − 8.63·45-s − 2.69·47-s − 6.90·49-s − 17.7·51-s + 4.15·53-s + 4.73·55-s + ⋯ |
| L(s) = 1 | − 1.96·3-s − 0.447·5-s + 0.117·7-s + 2.87·9-s − 1.42·11-s − 1.35·13-s + 0.880·15-s + 1.26·17-s + 1.21·19-s − 0.232·21-s − 1.56·23-s + 0.200·25-s − 3.70·27-s + 0.344·29-s + 0.845·31-s + 2.81·33-s − 0.0527·35-s − 0.583·37-s + 2.66·39-s + 1.53·41-s − 1.35·43-s − 1.28·45-s − 0.393·47-s − 0.986·49-s − 2.48·51-s + 0.570·53-s + 0.638·55-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
| \(L(1)\) |
\(\approx\) |
\(0.4008823330\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4008823330\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.41T + 3T^{2} \) |
| 7 | \( 1 - 0.311T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 + 4.87T + 13T^{2} \) |
| 17 | \( 1 - 5.19T + 17T^{2} \) |
| 19 | \( 1 - 5.28T + 19T^{2} \) |
| 23 | \( 1 + 7.51T + 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 - 4.70T + 31T^{2} \) |
| 37 | \( 1 + 3.55T + 37T^{2} \) |
| 41 | \( 1 - 9.83T + 41T^{2} \) |
| 43 | \( 1 + 8.90T + 43T^{2} \) |
| 47 | \( 1 + 2.69T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 - 1.04T + 59T^{2} \) |
| 61 | \( 1 + 0.751T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 1.76T + 79T^{2} \) |
| 83 | \( 1 - 17.7T + 83T^{2} \) |
| 89 | \( 1 + 18.6T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.64601084518791674499267666845, −7.20903685880963067560695878816, −6.28901251262739524971468372234, −5.64129311816774117998640241508, −5.05904659311170951915006733371, −4.72019973366147055974002773437, −3.71635167032068418808872108004, −2.63055424795870113191860967566, −1.42044227264847390249638023868, −0.36628763867131837430666108675,
0.36628763867131837430666108675, 1.42044227264847390249638023868, 2.63055424795870113191860967566, 3.71635167032068418808872108004, 4.72019973366147055974002773437, 5.05904659311170951915006733371, 5.64129311816774117998640241508, 6.28901251262739524971468372234, 7.20903685880963067560695878816, 7.64601084518791674499267666845
