| L(s) = 1 | + (−1.39 − 0.297i)2-s + (−1.19 − 1.25i)3-s + (0.0411 + 0.0183i)4-s + (1.76 − 1.37i)5-s + (1.30 + 2.10i)6-s + (1.92 − 3.32i)7-s + (2.26 + 1.64i)8-s + (−0.129 + 2.99i)9-s + (−2.87 + 1.39i)10-s + (−5.11 − 1.08i)11-s + (−0.0263 − 0.0733i)12-s + (3.28 − 0.699i)13-s + (−3.67 + 4.08i)14-s + (−3.83 − 0.560i)15-s + (−2.73 − 3.03i)16-s + (−3.90 − 2.84i)17-s + ⋯ |
| L(s) = 1 | + (−0.989 − 0.210i)2-s + (−0.691 − 0.722i)3-s + (0.0205 + 0.00915i)4-s + (0.788 − 0.614i)5-s + (0.532 + 0.859i)6-s + (0.725 − 1.25i)7-s + (0.799 + 0.580i)8-s + (−0.0431 + 0.999i)9-s + (−0.909 + 0.442i)10-s + (−1.54 − 0.327i)11-s + (−0.00760 − 0.0211i)12-s + (0.912 − 0.193i)13-s + (−0.982 + 1.09i)14-s + (−0.989 − 0.144i)15-s + (−0.683 − 0.759i)16-s + (−0.948 − 0.688i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.112941 - 0.510053i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.112941 - 0.510053i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.19 + 1.25i)T \) |
| 5 | \( 1 + (-1.76 + 1.37i)T \) |
| good | 2 | \( 1 + (1.39 + 0.297i)T + (1.82 + 0.813i)T^{2} \) |
| 7 | \( 1 + (-1.92 + 3.32i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (5.11 + 1.08i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-3.28 + 0.699i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (3.90 + 2.84i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.90 + 1.38i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (3.12 - 3.46i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.768 - 7.30i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.0679 - 0.646i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-0.315 + 0.970i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.66 + 0.779i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (2.34 - 4.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.24 + 11.8i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (-2.92 + 2.12i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-7.61 + 1.61i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-6.44 - 1.36i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.106 + 1.01i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (2.82 - 2.04i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.38 + 4.25i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.171 + 1.63i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-10.3 + 4.59i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (0.935 + 2.87i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.69 + 16.1i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−11.41700747571274211083860160739, −10.70389363727504070430695030969, −10.17506641008297948224112810401, −8.722565226831416797884340388041, −8.010794451067614702359593838486, −7.02285926513973464319514421285, −5.49939728095603011526236159599, −4.70792200155913329225811598654, −1.94552412664851846094799241911, −0.65958515409849120724929439821,
2.24247818473773957300388158690, 4.34718800878149595777024269790, 5.59057446422509405049481098539, 6.44597184261749273084714710654, 8.052543008159287841590109876960, 8.803664245696690895828708532076, 9.814000736436978934981547350543, 10.56584784787241131258471683249, 11.21951271273249625628543350844, 12.57494973433493222953359463926
