| L(s) = 1 | − 2.21·2-s + 2.92·4-s + 4.14·5-s + 3.37·7-s − 2.04·8-s − 9.19·10-s − 1.46·11-s − 2.37·13-s − 7.48·14-s − 1.30·16-s + 4.06·19-s + 12.1·20-s + 3.24·22-s + 8.37·23-s + 12.1·25-s + 5.26·26-s + 9.86·28-s + 6.84·29-s + 2.15·31-s + 6.98·32-s + 13.9·35-s − 2.16·37-s − 9.02·38-s − 8.47·40-s + 4.47·41-s + 5.91·43-s − 4.27·44-s + ⋯ |
| L(s) = 1 | − 1.56·2-s + 1.46·4-s + 1.85·5-s + 1.27·7-s − 0.723·8-s − 2.90·10-s − 0.441·11-s − 0.658·13-s − 2.00·14-s − 0.326·16-s + 0.933·19-s + 2.70·20-s + 0.692·22-s + 1.74·23-s + 2.43·25-s + 1.03·26-s + 1.86·28-s + 1.27·29-s + 0.386·31-s + 1.23·32-s + 2.36·35-s − 0.355·37-s − 1.46·38-s − 1.33·40-s + 0.698·41-s + 0.901·43-s − 0.644·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.840959117\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.840959117\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.21T + 2T^{2} \) |
| 5 | \( 1 - 4.14T + 5T^{2} \) |
| 7 | \( 1 - 3.37T + 7T^{2} \) |
| 11 | \( 1 + 1.46T + 11T^{2} \) |
| 13 | \( 1 + 2.37T + 13T^{2} \) |
| 19 | \( 1 - 4.06T + 19T^{2} \) |
| 23 | \( 1 - 8.37T + 23T^{2} \) |
| 29 | \( 1 - 6.84T + 29T^{2} \) |
| 31 | \( 1 - 2.15T + 31T^{2} \) |
| 37 | \( 1 + 2.16T + 37T^{2} \) |
| 41 | \( 1 - 4.47T + 41T^{2} \) |
| 43 | \( 1 - 5.91T + 43T^{2} \) |
| 47 | \( 1 + 7.02T + 47T^{2} \) |
| 53 | \( 1 + 1.32T + 53T^{2} \) |
| 59 | \( 1 - 4.38T + 59T^{2} \) |
| 61 | \( 1 + 6.74T + 61T^{2} \) |
| 67 | \( 1 - 0.452T + 67T^{2} \) |
| 71 | \( 1 - 2.98T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 2.26T + 79T^{2} \) |
| 83 | \( 1 - 9.83T + 83T^{2} \) |
| 89 | \( 1 + 8.05T + 89T^{2} \) |
| 97 | \( 1 - 9.72T + 97T^{2} \) |
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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
−7.960508196095272942354814842552, −7.33734446360838077631471380062, −6.71186635368909258954077333116, −5.90644924498673210728727962418, −4.97971720499914132017679615391, −4.81829064848835818610825708910, −2.88460347466262529308196591866, −2.37216753007345089488960535129, −1.48375301327598631883724476132, −0.972056710521769146527497044753,
0.972056710521769146527497044753, 1.48375301327598631883724476132, 2.37216753007345089488960535129, 2.88460347466262529308196591866, 4.81829064848835818610825708910, 4.97971720499914132017679615391, 5.90644924498673210728727962418, 6.71186635368909258954077333116, 7.33734446360838077631471380062, 7.960508196095272942354814842552
