| L(s) = 1 | + (0.275 + 1.38i)2-s + (−0.987 − 0.156i)3-s + (−1.84 + 0.763i)4-s + (−1.17 − 1.89i)5-s + (−0.0547 − 1.41i)6-s + (−1.39 + 1.39i)7-s + (−1.56 − 2.35i)8-s + (0.951 + 0.309i)9-s + (2.31 − 2.15i)10-s + (1.17 − 0.382i)11-s + (1.94 − 0.464i)12-s + (−2.63 − 5.16i)13-s + (−2.32 − 1.55i)14-s + (0.867 + 2.06i)15-s + (2.83 − 2.82i)16-s + (−0.916 − 5.78i)17-s + ⋯ |
| L(s) = 1 | + (0.194 + 0.980i)2-s + (−0.570 − 0.0903i)3-s + (−0.924 + 0.381i)4-s + (−0.527 − 0.849i)5-s + (−0.0223 − 0.576i)6-s + (−0.528 + 0.528i)7-s + (−0.554 − 0.832i)8-s + (0.317 + 0.103i)9-s + (0.730 − 0.682i)10-s + (0.355 − 0.115i)11-s + (0.561 − 0.134i)12-s + (−0.729 − 1.43i)13-s + (−0.620 − 0.415i)14-s + (0.224 + 0.532i)15-s + (0.708 − 0.705i)16-s + (−0.222 − 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.302973 - 0.260043i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.302973 - 0.260043i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.275 - 1.38i)T \) |
| 3 | \( 1 + (0.987 + 0.156i)T \) |
| 5 | \( 1 + (1.17 + 1.89i)T \) |
| good | 7 | \( 1 + (1.39 - 1.39i)T - 7iT^{2} \) |
| 11 | \( 1 + (-1.17 + 0.382i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (2.63 + 5.16i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.916 + 5.78i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (3.68 + 2.68i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (0.0552 - 0.108i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-3.15 - 4.34i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (6.30 - 8.67i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (3.90 - 1.98i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-1.88 + 5.81i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (6.87 + 6.87i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.721 - 4.55i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.00324 + 0.0205i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-2.84 + 8.75i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.217 - 0.668i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-7.05 + 1.11i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (4.63 + 6.38i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (4.73 + 2.41i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (-10.1 + 7.39i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (0.455 + 2.87i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-9.16 + 2.97i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (7.28 + 1.15i)T + (92.2 + 29.9i)T^{2} \) |
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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
−11.95581911039643190038420754178, −10.53619335021666167664151836489, −9.293315438909812196820706015607, −8.650790934861447439294463357254, −7.49452851860802117510849215404, −6.65611155287231563298769754451, −5.33640657169782871362644030479, −4.88311082733264153033537713000, −3.30211597743954959902052106188, −0.28963586788114492094737222902,
2.07927245505940994053641541336, 3.80518443976634303224470252112, 4.32073701793272867848090040675, 6.05991485321596241724554325508, 6.88105778214622267992443221774, 8.252224152847064484956280748758, 9.612599563037739898477645079362, 10.24723481015950439240806230709, 11.13095792568782806226497452068, 11.77375489589171685518193564880
