| L(s) = 1 | + (0.659 + 2.99i)2-s + (−2.18 − 2.05i)3-s + (−4.92 + 2.27i)4-s + (6.45 + 1.79i)5-s + (4.70 − 7.91i)6-s + (6.80 − 6.44i)7-s + (−2.64 − 3.47i)8-s + (0.577 + 8.98i)9-s + (−1.11 + 20.5i)10-s + (6.66 − 5.65i)11-s + (15.4 + 5.11i)12-s + (−5.87 + 1.97i)13-s + (23.8 + 16.1i)14-s + (−10.4 − 17.1i)15-s + (−5.36 + 6.31i)16-s + (−2.85 + 3.00i)17-s + ⋯ |
| L(s) = 1 | + (0.329 + 1.49i)2-s + (−0.729 − 0.684i)3-s + (−1.23 + 0.569i)4-s + (1.29 + 0.358i)5-s + (0.784 − 1.31i)6-s + (0.971 − 0.920i)7-s + (−0.330 − 0.434i)8-s + (0.0641 + 0.997i)9-s + (−0.111 + 2.05i)10-s + (0.605 − 0.514i)11-s + (1.28 + 0.426i)12-s + (−0.451 + 0.152i)13-s + (1.70 + 1.15i)14-s + (−0.696 − 1.14i)15-s + (−0.335 + 0.394i)16-s + (−0.167 + 0.177i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0659 - 0.997i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0659 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.33531 + 1.24999i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33531 + 1.24999i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.18 + 2.05i)T \) |
| 59 | \( 1 + (53.8 - 24.1i)T \) |
| good | 2 | \( 1 + (-0.659 - 2.99i)T + (-3.63 + 1.67i)T^{2} \) |
| 5 | \( 1 + (-6.45 - 1.79i)T + (21.4 + 12.8i)T^{2} \) |
| 7 | \( 1 + (-6.80 + 6.44i)T + (2.65 - 48.9i)T^{2} \) |
| 11 | \( 1 + (-6.66 + 5.65i)T + (19.5 - 119. i)T^{2} \) |
| 13 | \( 1 + (5.87 - 1.97i)T + (134. - 102. i)T^{2} \) |
| 17 | \( 1 + (2.85 - 3.00i)T + (-15.6 - 288. i)T^{2} \) |
| 19 | \( 1 + (-6.93 - 17.4i)T + (-262. + 248. i)T^{2} \) |
| 23 | \( 1 + (-3.92 - 36.0i)T + (-516. + 113. i)T^{2} \) |
| 29 | \( 1 + (-7.46 + 33.9i)T + (-763. - 353. i)T^{2} \) |
| 31 | \( 1 + (-16.2 + 40.8i)T + (-697. - 660. i)T^{2} \) |
| 37 | \( 1 + (1.16 + 0.883i)T + (366. + 1.31e3i)T^{2} \) |
| 41 | \( 1 + (2.65 - 24.3i)T + (-1.64e3 - 361. i)T^{2} \) |
| 43 | \( 1 + (-46.5 + 54.8i)T + (-299. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (38.9 - 10.8i)T + (1.89e3 - 1.13e3i)T^{2} \) |
| 53 | \( 1 + (41.9 - 2.27i)T + (2.79e3 - 303. i)T^{2} \) |
| 61 | \( 1 + (111. - 24.5i)T + (3.37e3 - 1.56e3i)T^{2} \) |
| 67 | \( 1 + (-7.82 + 5.95i)T + (1.20e3 - 4.32e3i)T^{2} \) |
| 71 | \( 1 + (-27.1 + 7.53i)T + (4.31e3 - 2.59e3i)T^{2} \) |
| 73 | \( 1 + (-19.8 + 29.2i)T + (-1.97e3 - 4.95e3i)T^{2} \) |
| 79 | \( 1 + (0.167 + 1.02i)T + (-5.91e3 + 1.99e3i)T^{2} \) |
| 83 | \( 1 + (83.8 - 44.4i)T + (3.86e3 - 5.70e3i)T^{2} \) |
| 89 | \( 1 + (-37.1 + 168. i)T + (-7.18e3 - 3.32e3i)T^{2} \) |
| 97 | \( 1 + (2.99 + 4.41i)T + (-3.48e3 + 8.74e3i)T^{2} \) |
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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
−13.28072155069776082713433257567, −11.73810138447338504515240596916, −10.79214691020627695218725843417, −9.592557629241679412169631136080, −7.990693770692088367174222993128, −7.34561845111510675157019374305, −6.21307585608345787317977588323, −5.69209866750759360180113554000, −4.46153969529186787829087897330, −1.65962676439884138685895443177,
1.41388209741634052886780311233, 2.73652187186631003767576659839, 4.70986729827247515648831931628, 5.08970483273245014530709705333, 6.55155188134592106063347872023, 8.918713047872499877164249350333, 9.482521849994127968094625324222, 10.47111988391497506714076807052, 11.20909568200257572348295809210, 12.23725255524912999378219926517
