| L(s) = 1 | + (0.456 + 1.33i)2-s + (−1.58 + 1.22i)4-s + (0.874 − 3.69i)5-s + (2.85 + 2.12i)7-s + (−2.35 − 1.55i)8-s + (5.33 − 0.515i)10-s + (0.253 + 0.268i)11-s + (1.86 − 0.937i)13-s + (−1.54 + 4.79i)14-s + (1.00 − 3.87i)16-s + (−2.09 − 5.76i)17-s + (1.03 − 2.83i)19-s + (3.12 + 6.91i)20-s + (−0.243 + 0.461i)22-s + (−3.75 − 5.04i)23-s + ⋯ |
| L(s) = 1 | + (0.323 + 0.946i)2-s + (−0.791 + 0.611i)4-s + (0.391 − 1.65i)5-s + (1.08 + 0.804i)7-s + (−0.834 − 0.551i)8-s + (1.68 − 0.162i)10-s + (0.0763 + 0.0809i)11-s + (0.517 − 0.260i)13-s + (−0.412 + 1.28i)14-s + (0.252 − 0.967i)16-s + (−0.508 − 1.39i)17-s + (0.236 − 0.650i)19-s + (0.699 + 1.54i)20-s + (−0.0519 + 0.0983i)22-s + (−0.783 − 1.05i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 972 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.168i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.99383 + 0.169634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.99383 + 0.169634i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.456 - 1.33i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.874 + 3.69i)T + (-4.46 - 2.24i)T^{2} \) |
| 7 | \( 1 + (-2.85 - 2.12i)T + (2.00 + 6.70i)T^{2} \) |
| 11 | \( 1 + (-0.253 - 0.268i)T + (-0.639 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.86 + 0.937i)T + (7.76 - 10.4i)T^{2} \) |
| 17 | \( 1 + (2.09 + 5.76i)T + (-13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (-1.03 + 2.83i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (3.75 + 5.04i)T + (-6.59 + 22.0i)T^{2} \) |
| 29 | \( 1 + (-3.89 - 5.92i)T + (-11.4 + 26.6i)T^{2} \) |
| 31 | \( 1 + (-2.55 - 1.10i)T + (21.2 + 22.5i)T^{2} \) |
| 37 | \( 1 + (0.159 + 0.133i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-10.2 - 0.596i)T + (40.7 + 4.75i)T^{2} \) |
| 43 | \( 1 + (-7.05 - 2.11i)T + (35.9 + 23.6i)T^{2} \) |
| 47 | \( 1 + (-2.93 - 6.81i)T + (-32.2 + 34.1i)T^{2} \) |
| 53 | \( 1 + (6.38 + 3.68i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.92 - 3.10i)T + (-3.43 - 58.9i)T^{2} \) |
| 61 | \( 1 + (1.15 + 0.134i)T + (59.3 + 14.0i)T^{2} \) |
| 67 | \( 1 + (-4.04 + 6.14i)T + (-26.5 - 61.5i)T^{2} \) |
| 71 | \( 1 + (-0.398 + 2.25i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.219 - 1.24i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.0782 + 0.00455i)T + (78.4 - 9.17i)T^{2} \) |
| 83 | \( 1 + (-0.294 - 5.05i)T + (-82.4 + 9.63i)T^{2} \) |
| 89 | \( 1 + (-8.23 + 1.45i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (12.4 - 2.94i)T + (86.6 - 43.5i)T^{2} \) |
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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
−9.442732416948495471651591278398, −9.038581404279429361925431183765, −8.370628183321503166909304447263, −7.72203226812897775771103382053, −6.44448971727880994307912957478, −5.53629448312729589275548160522, −4.85621742670117215081929689786, −4.40703315807122922852132901201, −2.58319337241567169564837365936, −0.949858879276335430217673523647,
1.51062472598162334396269495463, 2.44861936478601819822109840561, 3.73155007750050266837817145996, 4.24036389040199330802146462614, 5.76496243693940679437087724973, 6.32896796556805229933020077514, 7.55471424164814427122682458662, 8.289984865538280293999980320660, 9.556493831308835216092932744323, 10.32134272015728056581225691970
