| L(s) = 1 | + (0.753 + 1.19i)2-s + (2.39 − 1.20i)3-s + (−0.863 + 1.80i)4-s + (0.631 + 1.42i)5-s + (3.24 + 1.95i)6-s + (3.24 + 1.94i)7-s + (−2.80 + 0.326i)8-s + (2.48 − 3.35i)9-s + (−1.22 + 1.82i)10-s + (−3.18 − 2.48i)11-s + (0.106 + 5.35i)12-s + (−0.592 + 0.564i)13-s + (0.118 + 5.35i)14-s + (3.22 + 2.64i)15-s + (−2.50 − 3.11i)16-s + (−3.27 − 3.98i)17-s + ⋯ |
| L(s) = 1 | + (0.533 + 0.846i)2-s + (1.38 − 0.695i)3-s + (−0.431 + 0.901i)4-s + (0.282 + 0.637i)5-s + (1.32 + 0.798i)6-s + (1.22 + 0.735i)7-s + (−0.993 + 0.115i)8-s + (0.828 − 1.11i)9-s + (−0.388 + 0.578i)10-s + (−0.961 − 0.749i)11-s + (0.0307 + 1.54i)12-s + (−0.164 + 0.156i)13-s + (0.0317 + 1.43i)14-s + (0.833 + 0.683i)15-s + (−0.627 − 0.778i)16-s + (−0.793 − 0.966i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.46899 + 1.54862i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.46899 + 1.54862i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.753 - 1.19i)T \) |
| good | 3 | \( 1 + (-2.39 + 1.20i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-0.631 - 1.42i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-3.24 - 1.94i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (3.18 + 2.48i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (0.592 - 0.564i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (3.27 + 3.98i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-0.340 - 1.96i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (6.78 + 3.20i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-2.69 - 1.52i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (-3.42 + 0.681i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (-0.790 - 0.500i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (-2.12 - 2.34i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-1.46 - 4.42i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (1.28 + 0.685i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (-10.5 + 5.97i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (-6.99 + 7.35i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.182 - 2.47i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (4.22 + 3.64i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (10.2 + 1.52i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (9.61 - 5.76i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (-3.27 - 0.992i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (14.3 - 9.09i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (6.44 - 3.05i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (13.7 + 9.17i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.28115449767481117273469354845, −9.914827990808274938539215358337, −8.599820859115701240860853544026, −8.367178280374070313200021122705, −7.56249120307983627601556422030, −6.62320315396805213123788611256, −5.55970289665525748086186374532, −4.41185984402859048104504180333, −2.87733776614298720399215865666, −2.32500719010214050702164241896,
1.67707563484853495551779512909, 2.63169753336510859474129572797, 4.13077027048731241727070136261, 4.50285503206894569988203712781, 5.56320567249480189622117799022, 7.43973779281038039165726163636, 8.372602052856948810450087726245, 9.007331203545061624979368568620, 10.14735191489547504777953451392, 10.39910928832977567439288265572
