| L(s) = 1 | + (−1.99 − 1.72i)2-s + (0.319 + 1.70i)3-s + (0.702 + 4.88i)4-s + (1.01 + 2.21i)5-s + (2.29 − 3.93i)6-s + (−0.333 − 1.13i)7-s + (4.17 − 6.49i)8-s + (−2.79 + 1.08i)9-s + (1.80 − 6.16i)10-s + (1.91 + 2.20i)11-s + (−8.08 + 2.75i)12-s + (2.12 + 0.624i)13-s + (−1.29 + 2.83i)14-s + (−3.45 + 2.43i)15-s + (−10.0 + 2.95i)16-s + (0.460 − 3.20i)17-s + ⋯ |
| L(s) = 1 | + (−1.40 − 1.21i)2-s + (0.184 + 0.982i)3-s + (0.351 + 2.44i)4-s + (0.452 + 0.991i)5-s + (0.938 − 1.60i)6-s + (−0.126 − 0.429i)7-s + (1.47 − 2.29i)8-s + (−0.931 + 0.363i)9-s + (0.571 − 1.94i)10-s + (0.576 + 0.665i)11-s + (−2.33 + 0.796i)12-s + (0.589 + 0.173i)13-s + (−0.346 + 0.757i)14-s + (−0.891 + 0.628i)15-s + (−2.51 + 0.737i)16-s + (0.111 − 0.776i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.530877 + 0.0685150i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.530877 + 0.0685150i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.319 - 1.70i)T \) |
| 23 | \( 1 + (2.40 + 4.15i)T \) |
| good | 2 | \( 1 + (1.99 + 1.72i)T + (0.284 + 1.97i)T^{2} \) |
| 5 | \( 1 + (-1.01 - 2.21i)T + (-3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (0.333 + 1.13i)T + (-5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-1.91 - 2.20i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.12 - 0.624i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.460 + 3.20i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (2.06 - 0.297i)T + (18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-3.98 - 0.573i)T + (27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (3.31 + 2.12i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-9.07 - 4.14i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (1.04 - 0.476i)T + (26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (5.65 + 8.79i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 3.92iT - 47T^{2} \) |
| 53 | \( 1 + (1.17 - 0.345i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (0.856 - 2.91i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (0.408 - 0.634i)T + (-25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-3.21 - 2.78i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-5.67 - 4.91i)T + (10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (0.227 + 1.58i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (-0.756 + 2.57i)T + (-66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (2.64 - 5.79i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (4.54 - 2.92i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (11.4 - 5.23i)T + (63.5 - 73.3i)T^{2} \) |
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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
−14.85430039708684732179381276842, −13.64845695103020981018455175049, −12.00282798262490020463198705293, −10.96637266102346556024488905541, −10.23510409426286604709664552790, −9.527770310936841407330249528788, −8.365427373035244220455623729934, −6.81058067381258651305613419039, −3.97172198974463923312610295125, −2.55000771174214931495786828484,
1.35837991875808889062374246077, 5.68495136111454977311677888009, 6.40841064010277529781813026608, 7.952069840670277093326683888851, 8.688657570275387490533585653827, 9.483949244531358244215277369130, 11.14749165535252525394014230109, 12.70103130021910005474283856274, 13.85341904911169636462873508220, 14.91970239113714707797767702570
