| L(s) = 1 | + (−2.45 − 0.214i)2-s + (−0.566 − 1.63i)3-s + (4.00 + 0.706i)4-s + (1.21 − 1.87i)5-s + (1.03 + 4.13i)6-s + (1.02 + 0.719i)7-s + (−4.92 − 1.32i)8-s + (−2.35 + 1.85i)9-s + (−3.37 + 4.35i)10-s + (0.955 − 2.62i)11-s + (−1.11 − 6.96i)12-s + (−0.519 − 5.93i)13-s + (−2.36 − 1.98i)14-s + (−3.76 − 0.916i)15-s + (4.16 + 1.51i)16-s + (−5.37 + 1.44i)17-s + ⋯ |
| L(s) = 1 | + (−1.73 − 0.151i)2-s + (−0.327 − 0.944i)3-s + (2.00 + 0.353i)4-s + (0.541 − 0.840i)5-s + (0.424 + 1.68i)6-s + (0.388 + 0.271i)7-s + (−1.74 − 0.466i)8-s + (−0.785 + 0.618i)9-s + (−1.06 + 1.37i)10-s + (0.287 − 0.791i)11-s + (−0.322 − 2.00i)12-s + (−0.143 − 1.64i)13-s + (−0.632 − 0.530i)14-s + (−0.971 − 0.236i)15-s + (1.04 + 0.378i)16-s + (−1.30 + 0.349i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.214461 - 0.402226i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.214461 - 0.402226i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.566 + 1.63i)T \) |
| 5 | \( 1 + (-1.21 + 1.87i)T \) |
| good | 2 | \( 1 + (2.45 + 0.214i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-1.02 - 0.719i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.955 + 2.62i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.519 + 5.93i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (5.37 - 1.44i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.75 - 1.01i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.67 - 3.82i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-2.13 + 1.79i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.860 + 4.87i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.354 - 1.32i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (0.207 - 0.247i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.79 - 8.13i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (0.213 - 0.305i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-7.96 + 7.96i)T - 53iT^{2} \) |
| 59 | \( 1 + (-2.97 + 1.08i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.0275 + 0.156i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-8.09 + 0.707i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-7.01 - 4.04i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.41 - 5.29i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-10.3 - 12.2i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.151 - 1.73i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-6.44 - 11.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.21 - 1.49i)T + (62.3 - 74.3i)T^{2} \) |
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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
−12.72717133912378456137758374963, −11.54809294132148595249942193980, −10.81523338698370639410953757019, −9.594554471860065647887462140095, −8.401464970808025142724474325379, −8.104011881071867147086595040253, −6.59804049162858367959116206643, −5.48633029161955114962589351737, −2.32496661448907597491405339987, −0.853254042467374960457562551723,
2.22407025175709470904981076097, 4.51678769149198436805758806198, 6.51506375951981528157579569236, 7.07014062549293043451023179416, 8.859991545832500326832818730090, 9.335900888064369837561583650697, 10.44918666829119613644119303099, 10.95016905202443797407718893776, 11.89098357585304244802214475366, 13.98412960909930868992622257914
