| L(s) = 1 | + (−0.691 − 1.23i)2-s + (1.38 − 1.04i)3-s + (−1.04 + 1.70i)4-s + (1.90 + 2.27i)5-s + (−2.24 − 0.983i)6-s + (1.28 − 0.466i)7-s + (2.82 + 0.109i)8-s + (0.820 − 2.88i)9-s + (1.48 − 3.92i)10-s + (−2.78 + 3.31i)11-s + (0.337 + 3.44i)12-s + (4.65 − 0.820i)13-s + (−1.46 − 1.25i)14-s + (5.01 + 1.15i)15-s + (−1.81 − 3.56i)16-s + (0.356 + 0.618i)17-s + ⋯ |
| L(s) = 1 | + (−0.488 − 0.872i)2-s + (0.797 − 0.602i)3-s + (−0.522 + 0.852i)4-s + (0.853 + 1.01i)5-s + (−0.915 − 0.401i)6-s + (0.484 − 0.176i)7-s + (0.999 + 0.0386i)8-s + (0.273 − 0.961i)9-s + (0.470 − 1.24i)10-s + (−0.839 + 1.00i)11-s + (0.0973 + 0.995i)12-s + (1.29 − 0.227i)13-s + (−0.390 − 0.336i)14-s + (1.29 + 0.297i)15-s + (−0.454 − 0.890i)16-s + (0.0865 + 0.149i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19550 - 0.613765i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19550 - 0.613765i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.691 + 1.23i)T \) |
| 3 | \( 1 + (-1.38 + 1.04i)T \) |
| good | 5 | \( 1 + (-1.90 - 2.27i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-1.28 + 0.466i)T + (5.36 - 4.49i)T^{2} \) |
| 11 | \( 1 + (2.78 - 3.31i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-4.65 + 0.820i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.356 - 0.618i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.43 + 3.71i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.17 + 1.51i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (6.69 + 1.18i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-5.02 - 1.82i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.547 + 0.315i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.356 - 2.02i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.43 + 2.90i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-5.10 + 1.85i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 11.4iT - 53T^{2} \) |
| 59 | \( 1 + (0.803 + 0.957i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.216 - 0.595i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (14.0 - 2.47i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.414 + 0.717i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.29 + 12.6i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.22 - 6.96i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.29 - 0.227i)T + (77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.66 + 4.61i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.52 + 1.27i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.25300127767153842103940838369, −10.87458087061174256362355424504, −10.39045068123499435939287325509, −9.303187569758799127545188002955, −8.288384411310511932059706108598, −7.39809140146770534237055001134, −6.24718454558919045180701650337, −4.21461170500486331011616569205, −2.73994463386483406807145710994, −1.86627361150370726022256439971,
1.79151834961587802677877758685, 4.07476749562733212456117344617, 5.33092143142541678694721767862, 6.07275824807447837857588713565, 8.008662080163331518832916104112, 8.459939954664336527087521575114, 9.218294963585631006085415038873, 10.21519874588575419022021781457, 11.07111338691674904788700113601, 13.03367516536115514069014270920
