| L(s) = 1 | − 1.83·2-s − 2.83·3-s + 1.36·4-s + 5.19·6-s + 1.16·8-s + 5.03·9-s − 6.39·11-s − 3.86·12-s − 3.86·13-s − 4.86·16-s + 2.36·17-s − 9.23·18-s + 8.03·19-s + 11.7·22-s + 1.56·23-s − 3.30·24-s − 5·25-s + 7.09·26-s − 5.76·27-s − 3.66·29-s + 6.79·31-s + 6.59·32-s + 18.1·33-s − 4.33·34-s + 6.86·36-s − 9.86·37-s − 14.7·38-s + ⋯ |
| L(s) = 1 | − 1.29·2-s − 1.63·3-s + 0.682·4-s + 2.12·6-s + 0.412·8-s + 1.67·9-s − 1.92·11-s − 1.11·12-s − 1.07·13-s − 1.21·16-s + 0.573·17-s − 2.17·18-s + 1.84·19-s + 2.50·22-s + 0.325·23-s − 0.674·24-s − 25-s + 1.39·26-s − 1.10·27-s − 0.681·29-s + 1.22·31-s + 1.16·32-s + 3.15·33-s − 0.743·34-s + 1.14·36-s − 1.62·37-s − 2.39·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1814754689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1814754689\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.83T + 2T^{2} \) |
| 3 | \( 1 + 2.83T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 6.39T + 11T^{2} \) |
| 13 | \( 1 + 3.86T + 13T^{2} \) |
| 17 | \( 1 - 2.36T + 17T^{2} \) |
| 19 | \( 1 - 8.03T + 19T^{2} \) |
| 23 | \( 1 - 1.56T + 23T^{2} \) |
| 29 | \( 1 + 3.66T + 29T^{2} \) |
| 31 | \( 1 - 6.79T + 31T^{2} \) |
| 37 | \( 1 + 9.86T + 37T^{2} \) |
| 43 | \( 1 + 0.364T + 43T^{2} \) |
| 47 | \( 1 + 8.59T + 47T^{2} \) |
| 53 | \( 1 + 6.39T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 + 5.85T + 61T^{2} \) |
| 67 | \( 1 + 8.46T + 67T^{2} \) |
| 71 | \( 1 - 4.06T + 71T^{2} \) |
| 73 | \( 1 - 7.73T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 4.72T + 83T^{2} \) |
| 89 | \( 1 + 3.86T + 89T^{2} \) |
| 97 | \( 1 - 0.768T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.575028041170615526143126212128, −8.188761097932576417423086056770, −7.56844208204256961303118428006, −7.13082643109043463860779240304, −5.95436469359043940552632019504, −5.11025189147687812332511829522, −4.82350233916463722790187803215, −3.04375578964435953909760925159, −1.64348191700167938959836211436, −0.37681834620498450799425196019,
0.37681834620498450799425196019, 1.64348191700167938959836211436, 3.04375578964435953909760925159, 4.82350233916463722790187803215, 5.11025189147687812332511829522, 5.95436469359043940552632019504, 7.13082643109043463860779240304, 7.56844208204256961303118428006, 8.188761097932576417423086056770, 9.575028041170615526143126212128
