L(s) = 1 | + (−0.459 + 1.33i)2-s + i·3-s + (−1.57 − 1.22i)4-s + 2.04i·5-s + (−1.33 − 0.459i)6-s − 4.47·7-s + (2.36 − 1.54i)8-s − 9-s + (−2.74 − 0.941i)10-s − 3.86i·11-s + (1.22 − 1.57i)12-s − 5.65i·13-s + (2.05 − 5.98i)14-s − 2.04·15-s + (0.977 + 3.87i)16-s − 0.763·17-s + ⋯ |
L(s) = 1 | + (−0.324 + 0.945i)2-s + 0.577i·3-s + (−0.788 − 0.614i)4-s + 0.916i·5-s + (−0.546 − 0.187i)6-s − 1.69·7-s + (0.837 − 0.546i)8-s − 0.333·9-s + (−0.866 − 0.297i)10-s − 1.16i·11-s + (0.354 − 0.455i)12-s − 1.56i·13-s + (0.549 − 1.59i)14-s − 0.529·15-s + (0.244 + 0.969i)16-s − 0.185·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.184554 - 0.0999892i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.184554 - 0.0999892i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.459 - 1.33i)T \) |
| 3 | \( 1 - iT \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 2.04iT - 5T^{2} \) |
| 7 | \( 1 + 4.47T + 7T^{2} \) |
| 11 | \( 1 + 3.86iT - 11T^{2} \) |
| 13 | \( 1 + 5.65iT - 13T^{2} \) |
| 17 | \( 1 + 0.763T + 17T^{2} \) |
| 19 | \( 1 - 5.88iT - 19T^{2} \) |
| 29 | \( 1 + 3.85iT - 29T^{2} \) |
| 31 | \( 1 + 9.20T + 31T^{2} \) |
| 37 | \( 1 + 3.75iT - 37T^{2} \) |
| 41 | \( 1 + 3.32T + 41T^{2} \) |
| 43 | \( 1 + 1.84iT - 43T^{2} \) |
| 47 | \( 1 + 6.55T + 47T^{2} \) |
| 53 | \( 1 + 2.26iT - 53T^{2} \) |
| 59 | \( 1 + 10.6iT - 59T^{2} \) |
| 61 | \( 1 - 9.32iT - 61T^{2} \) |
| 67 | \( 1 + 3.64iT - 67T^{2} \) |
| 71 | \( 1 + 6.14T + 71T^{2} \) |
| 73 | \( 1 + 2.21T + 73T^{2} \) |
| 79 | \( 1 + 5.59T + 79T^{2} \) |
| 83 | \( 1 + 3.14iT - 83T^{2} \) |
| 89 | \( 1 + 0.345T + 89T^{2} \) |
| 97 | \( 1 + 19.4T + 97T^{2} \) |
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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
−10.33999155512340375394892035599, −9.848835112306715296089139461877, −8.877243410075526637580090215104, −7.950056313102243021041856870579, −6.92964434631457904005950359495, −6.01466962079735728236414729324, −5.55526859945179717254344886041, −3.71379502037302906938324642286, −3.12684761322979933516167243857, −0.13424595055194063385593298722,
1.57986828347929004910421630560, 2.82521362204917698870964097451, 4.11776436974281842527046039202, 5.07234594952395635662197569397, 6.69393926340965436012732398928, 7.22612649054884590050596859210, 8.759945850289063597569671856439, 9.223366085999939567303195962024, 9.807629035379634497734471916184, 11.01135815770940707446846128058