| L(s) = 1 | − 2.70·2-s + 5.30·4-s − 3.42·5-s − 1.82·7-s − 8.93·8-s + 9.24·10-s + 0.926·11-s − 4.26·13-s + 4.93·14-s + 13.5·16-s − 3.66·17-s + 4.08·19-s − 18.1·20-s − 2.50·22-s − 3.32·23-s + 6.70·25-s + 11.5·26-s − 9.68·28-s − 8.34·29-s − 3.46·31-s − 18.7·32-s + 9.90·34-s + 6.24·35-s − 9.91·37-s − 11.0·38-s + 30.5·40-s + 7.41·41-s + ⋯ |
| L(s) = 1 | − 1.91·2-s + 2.65·4-s − 1.52·5-s − 0.689·7-s − 3.15·8-s + 2.92·10-s + 0.279·11-s − 1.18·13-s + 1.31·14-s + 3.38·16-s − 0.888·17-s + 0.936·19-s − 4.05·20-s − 0.533·22-s − 0.694·23-s + 1.34·25-s + 2.26·26-s − 1.82·28-s − 1.55·29-s − 0.622·31-s − 3.31·32-s + 1.69·34-s + 1.05·35-s − 1.63·37-s − 1.79·38-s + 4.83·40-s + 1.15·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01006652501\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01006652501\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 5 | \( 1 + 3.42T + 5T^{2} \) |
| 7 | \( 1 + 1.82T + 7T^{2} \) |
| 11 | \( 1 - 0.926T + 11T^{2} \) |
| 13 | \( 1 + 4.26T + 13T^{2} \) |
| 17 | \( 1 + 3.66T + 17T^{2} \) |
| 19 | \( 1 - 4.08T + 19T^{2} \) |
| 23 | \( 1 + 3.32T + 23T^{2} \) |
| 29 | \( 1 + 8.34T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 + 9.91T + 37T^{2} \) |
| 41 | \( 1 - 7.41T + 41T^{2} \) |
| 43 | \( 1 - 4.79T + 43T^{2} \) |
| 47 | \( 1 - 1.83T + 47T^{2} \) |
| 53 | \( 1 + 5.01T + 53T^{2} \) |
| 59 | \( 1 + 14.7T + 59T^{2} \) |
| 61 | \( 1 + 1.21T + 61T^{2} \) |
| 67 | \( 1 - 0.101T + 67T^{2} \) |
| 71 | \( 1 - 4.13T + 71T^{2} \) |
| 73 | \( 1 + 4.05T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 8.34T + 83T^{2} \) |
| 89 | \( 1 + 6.88T + 89T^{2} \) |
| 97 | \( 1 - 7.92T + 97T^{2} \) |
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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
−7.894245645044628482405513012257, −7.60419729949666118598152160066, −7.10016143077533929184819896922, −6.42681307930165017852815667221, −5.43842011435371110085248680737, −4.19603799201202552324534344081, −3.37035563449779653638010267631, −2.58744740908244401434534257959, −1.52334975661345657525630854040, −0.06957926293439385030971263397,
0.06957926293439385030971263397, 1.52334975661345657525630854040, 2.58744740908244401434534257959, 3.37035563449779653638010267631, 4.19603799201202552324534344081, 5.43842011435371110085248680737, 6.42681307930165017852815667221, 7.10016143077533929184819896922, 7.60419729949666118598152160066, 7.894245645044628482405513012257
