L(s) = 1 | + (2.01 − 1.16i)2-s + (1.21 + 1.23i)3-s + (1.71 − 2.97i)4-s + (3.89 + 1.07i)6-s + (−1.11 − 2.39i)7-s − 3.33i·8-s + (−0.0404 + 2.99i)9-s + (2.42 + 1.39i)11-s + (5.75 − 1.50i)12-s − 3.20i·13-s + (−5.04 − 3.53i)14-s + (−0.459 − 0.795i)16-s + (0.440 − 0.763i)17-s + (3.41 + 6.10i)18-s + (1.90 − 1.09i)19-s + ⋯ |
L(s) = 1 | + (1.42 − 0.824i)2-s + (0.702 + 0.711i)3-s + (0.858 − 1.48i)4-s + (1.58 + 0.437i)6-s + (−0.422 − 0.906i)7-s − 1.18i·8-s + (−0.0134 + 0.999i)9-s + (0.729 + 0.421i)11-s + (1.66 − 0.433i)12-s − 0.888i·13-s + (−1.34 − 0.946i)14-s + (−0.114 − 0.198i)16-s + (0.106 − 0.185i)17-s + (0.804 + 1.43i)18-s + (0.436 − 0.251i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 + 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.40140 - 1.34505i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.40140 - 1.34505i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.21 - 1.23i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.11 + 2.39i)T \) |
good | 2 | \( 1 + (-2.01 + 1.16i)T + (1 - 1.73i)T^{2} \) |
| 11 | \( 1 + (-2.42 - 1.39i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.20iT - 13T^{2} \) |
| 17 | \( 1 + (-0.440 + 0.763i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.90 + 1.09i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.53 - 3.77i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.15iT - 29T^{2} \) |
| 31 | \( 1 + (7.62 + 4.40i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.203 - 0.352i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.55T + 41T^{2} \) |
| 43 | \( 1 - 0.118T + 43T^{2} \) |
| 47 | \( 1 + (-1.31 - 2.27i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.46 + 3.73i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.04 + 3.55i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-10.7 + 6.17i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.802 - 1.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6.25iT - 71T^{2} \) |
| 73 | \( 1 + (0.192 + 0.110i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.56 - 2.71i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 0.666T + 83T^{2} \) |
| 89 | \( 1 + (0.437 + 0.757i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6.37iT - 97T^{2} \) |
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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
−10.83286579760454785677416215860, −10.08125555006210336181483301414, −9.414471667366599764848604032231, −8.036947833337939205413804122490, −6.98119852407194423962424290774, −5.64604633060073358105100142228, −4.75460017465364916091826136862, −3.71325994148081488947575299607, −3.30990958103214532200102071758, −1.82390219463652049738086499245,
2.11544204082197914730367169990, 3.36183030853606177822522356879, 4.17276801258913943494544609789, 5.64176418136357822076882960314, 6.31618678553586654654656413473, 6.99204342621726132385087411200, 8.073721351025989885880951879086, 8.890244417975972970256603313174, 9.844114600108177439813753462418, 11.72981885904314270783140672303