| L(s) = 1 | + (2.99 + 0.118i)3-s + (3.29 + 0.580i)5-s + (−3.84 + 3.22i)7-s + (8.97 + 0.712i)9-s + (−7.73 + 1.36i)11-s + (13.9 − 5.07i)13-s + (9.79 + 2.13i)15-s + (18.4 + 10.6i)17-s + (12.0 + 20.8i)19-s + (−11.9 + 9.21i)21-s + (13.5 − 16.1i)23-s + (−12.9 − 4.72i)25-s + (26.8 + 3.20i)27-s + (−10.7 + 29.5i)29-s + (−3.12 − 2.62i)31-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0396i)3-s + (0.658 + 0.116i)5-s + (−0.549 + 0.460i)7-s + (0.996 + 0.0792i)9-s + (−0.703 + 0.124i)11-s + (1.07 − 0.390i)13-s + (0.653 + 0.142i)15-s + (1.08 + 0.625i)17-s + (0.633 + 1.09i)19-s + (−0.567 + 0.438i)21-s + (0.588 − 0.701i)23-s + (−0.519 − 0.189i)25-s + (0.992 + 0.118i)27-s + (−0.370 + 1.01i)29-s + (−0.100 − 0.0846i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.894 - 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.62824 + 0.619299i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.62824 + 0.619299i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.99 - 0.118i)T \) |
| good | 5 | \( 1 + (-3.29 - 0.580i)T + (23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (3.84 - 3.22i)T + (8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (7.73 - 1.36i)T + (113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-13.9 + 5.07i)T + (129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-18.4 - 10.6i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-12.0 - 20.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-13.5 + 16.1i)T + (-91.8 - 520. i)T^{2} \) |
| 29 | \( 1 + (10.7 - 29.5i)T + (-644. - 540. i)T^{2} \) |
| 31 | \( 1 + (3.12 + 2.62i)T + (166. + 946. i)T^{2} \) |
| 37 | \( 1 + (-20.6 + 35.8i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (1.46 + 4.01i)T + (-1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-2.94 - 16.6i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (42.4 + 50.5i)T + (-383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + 63.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-75.1 - 13.2i)T + (3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (76.5 - 64.2i)T + (646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (83.0 - 30.2i)T + (3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (4.34 + 2.50i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (19.2 + 33.3i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-35.9 - 13.0i)T + (4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-48.8 + 134. i)T + (-5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-75.2 + 43.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-28.6 - 162. i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−10.61299479014407295648216044987, −10.07449689353116717619861274087, −9.202316915558478097405457589846, −8.323046691434043786291548322756, −7.52039247315137017768714933910, −6.21797439721391789516456051168, −5.40150575353036754096553745428, −3.75116265324749420010571366985, −2.89252028512920881704308246025, −1.57667324688569479481163540424,
1.21749776891564682891973062072, 2.74417247084733398373574624540, 3.65589479958037241146559057159, 5.04209319875263236907104016941, 6.24282432330240604885170199823, 7.32401628510331378338948250341, 8.093415715442036115564418061634, 9.361646771325945356975839302037, 9.614181851754091314677596805692, 10.69676476317906797059304035184
